diff --git "a/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/load_file.txt" "b/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/7dAzT4oBgHgl3EQf-f5K/content/tmp_files/load_file.txt" @@ -0,0 +1,1136 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf,len=1135 +page_content='HYPERBOLIC AND SATELLITE LORENZ LINKS OBTAINED BY TWISTING THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A Lorenz link is equivalent to a T-link, which is a positive braid built by concatenating torus braids of increasing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When each torus braid except the largest is obtained by full twists, then the T-link can be described as the Dehn filling of a parent link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this paper, we completely classify when such parent links are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This gives a classification of the geometry of T-links obtained by full twists when the amount of twisting is large, although the bound on the number of required twists is not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We also present effective results on hyperbolicity for two families of T-links obtained by twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, we identify families of satellite T-links obtained by half-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Introduction Lorenz links are the closed periodic orbits of a system of equations in- vestigated by Lorenz in the 1960s [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' They exhibit interesting dynamics that has led to significant further investigation over the years, in the fields of dynamics, geometry, and topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' see for example [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These links can be described as links on an embedded branched surface in R3, called the Lorenz template, due to work of Guckenheimer and Williams [12], and Tucker [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Birman and Williams were the first to investigate Lorenz links through the lens of knot theory, in the 1980s [2], and the first to show such links are closed positive braids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Birman and Kofman [1] showed that Lorenz links are equivalent to T-links, which are positive braids with a particular form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' see Section 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus techniques from braid theory can be brought to bear upon Lorenz links via T-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We are interested in the complement of these links, and in particular their geometrisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thurston showed in the 1980s that all knots in the 3-sphere are either torus knots, satellite, or hyperbolic [20], and we refer to this as the knot’s geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The geometric type of Lorenz links has been considered since work of Birman and Williams in the 1980s [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' They showed that all torus knots are Lorenz knots, and satellites obtained as certain cables of Lorenz knots are Lorenz knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hyperbolic geometry has been considered by Gomes, Franco, and Silva [10, 11], who proved hyperbolicity of Lorenz links satisfying certain conditions based on the Lorenz template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Satellite links have received additional attention, by El Rifai [7], de Paiva [4], and de Paiva and Purcell [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='01934v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='GT] 5 Jan 2023 2 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL In spite of this work, there remains no systematic way of determining whether a Lorenz link is hyperbolic, toroidal, or satellite using its description either on the Lorenz template, or as a closed braid in the form of a T-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These descriptions uniquely determine a link, and hence uniquely determine its geometric type, so it is natural to ask for a simple description of geometric type based on the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We focus on T-links in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The paper [6] begins a classification of the geometry of T-links, by finding examples that are satellite and also by identifying certain “parent links”, which give classes of T-links under Dehn filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' While the work in that paper finds examples of satellite and hyperbolic links, it is incomplete for two reasons: (1) First, the hyperbolic geometry of the parent links is used to determine geometry of T-links for many examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But the classification of the hyperbolic geometry of the parent links is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (2) Second, because the results are obtained by Dehn filling, they apply only to links that admit full twists as T-link parameters, which are not required for general T-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this paper, we extend the classification of geometry of T-links as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' First, we complete the classification of item (1) above: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 completely classifies when parent links of fully twisted T-links are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This can be seen as an extension of work of Lee [16, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7], who proved a similar result for twisted torus knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Positive twisted torus knots are T-links with only one additional torus braid besides the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lee’s result essentially proves Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 in the case of only one additional link component in the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Our result applies to any number of additional link components in the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 leads to new infinite families of hyperbolic T-links, determined only by parameters in a braid describing the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Fix relatively prime integers q < p, and let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an be integers less than p and increasing in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There exists B ≫ 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the T-link obtained from the (p, q)-torus knot by full twisting at least B times in regions with a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an strands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This Lorenz link is hyperbolic if and only if either all ai < q, or there is ai > q that is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The T-links of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1 must be obtained by full twisting, and we currently do not have a concrete, universal bound on the number of full twists that are required in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' this is the constant B in the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In Section 4 we improve this: We present two theorems that guarantee hyperbolicity of T-links with full twists, given only their parameters, where the bounds on numbers of full twists required are explicit and relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The results are Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It seems much more difficult to address item (2), especially in the hyperbolic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There are some partial results known, for example by de Paiva for torus knots [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this paper, we give more results in the satellite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We LORENZ LINKS OBTAINED BY TWISTING 3 extend the results on satellite knots, requiring full twists in [6], to families of T-links with both full twists and half twists, which gives many more families in a very natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For q < p integers, let K be a T-link obtained from the (p, q)-torus link by half-twisting in circles encircling less than q strands, or encircling multiples of q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then S3 − K is satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The precise statement is Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This work was partially supported by the Aus- tralian Research Council, grant DP210103136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Results on braids This section reviews results on braids that will be used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As usual, let σi be the standard generator of the braid group, giving a positive crossing between the i-th and (i + 1)-th strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For 1 < p, q, define the (p, q)-torus braid as: (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q Note that within the braid group on p strands, its closure is the torus link T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When p, q are coprime, this is a torus knot, but we will not always restrict to coprime p and q unless specifically stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We will also consider such braids within larger braid groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When r < p, the (r, s) braid within the braid group on p strands is still defined to be (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)s, but now note this has p − r strands with no crossings lying to the right of the braid, viewing the braid arranged from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , rk and si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sk be integers such that 2 ≤ r1 < · · · < rk, and si > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The T-link T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rk, sk)) is defined to be the closure of the braid (σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr1−1)s1(σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr2−1)s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrk−1)sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rk, sk)) is obtained by concatenating the braids (ri, si) within the braid group on rk strands, and then taking the closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Taking closures of torus braids and related braids allows additional sym- metries and restrictions on the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For example, we will use the following standard result on torus knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let 1 < p, q be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the torus link T(p, q) is equiva- lent to the torus link T(q, p) via a homeomorphism of S3 fixing the Heegaard torus containing T(p, q) and switching the two solid tori bounded by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1 is well known, and appears in many knot theory texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We visualise the proof in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The next result generalises [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There the result only holds when each si is a multiple of ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Here we extend more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 4 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL p q p q Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The equivalence of T(p, q) and T(q, p) is given by rotating 180◦ in the diagonal axis shown for the Heegaard torus for S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This exchanges the solid tori in the standard genus-1 Heegaard splitting for S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let 0 < r1 < · · · < ri−1 < q < ri+1 < · · · < rn < p be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, for k > 0, the T-link K = T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (ri−1, si−1), (q, qk), (ri+1, si+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rn, sn), (p, q)) is equivalent to the T-link K′ = T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (ri−1, si−1), (ri+1, si+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rn, sn), (p + qk, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 allows us to assume there are no full twists on q strands in a T-link of the form T(· · · , (p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The braid (q, qk) is obtained by performing k full twists on q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We know that these full twists commute in the braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus in the braid representing K, we may isotope (q, qk) to the top of the braid, leaving the rest of the braid unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Now perform the isotopy of K of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1, switching p and q in the (p, q)-torus link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The rotation in the diagonal shown in Figure 1 takes the (vertical) braids (r1, s1) ∗ · · · ∗ (rn, sn) to inverted braids, forming a tangle in the horizontal direction on a quadrilateral representing the projection torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (The form of this tangle is not important for the argument here, but more details can be found in [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=') The result is a link of the form T(q, p) with a tangle along the horizontal p-strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The first such tangle is the braid (q, qk), which is unchanged by this isotopy because it is a full twist (see, for example, Birman and Kofman [1, Corollary 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link diagram is formed by the braid (q, p) followed by (q, qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These two braids can be combined to form the braid (q, p + qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Now apply the inverse of the isotopy of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This changes the link from T(q, p + qk) with tangles along the p horizontal strands to a link of the form T(p + qk, p) with these tangles returned to their form as braids (r1, s1) ∗ · · · ∗ (rn, sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The result is the link K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q, and r be positive integers with 0 < q ≤ r < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the (p, q) torus link, which is the closure of the braid on p strands given by (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There is an ambient isotopy of S3 taking this to the LORENZ LINKS OBTAINED BY TWISTING 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Illustration of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 in the case that q = 2, r = 4, p = 7, for an arbitrary tangle shown as a gray box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The left-most picture shows the original link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The (r + 1)-st strand, shown in blue, can be pulled tight beneath the diagram, resulting in the middle picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The right-most picture shows the result after isotoping strands (r + 1) to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' closure of the braid on r strands given by (σr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−q+1)p−r(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Moreover, an ambient isotopy realising the equivalence fixes the portion of the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q corresponding to the r left-most strands at the top the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, we may replace a neighbourhood of these strands above the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q with any tangle τ on r strands, and we find that the resulting link is ambient isotopic to the closure of the link obtained by concatenating the braid on r strands (σr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−q+1)p−r, with τ, and then with (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because r ≥ q, the (r + 1)-st strand at the top of the braid only runs under the q overcrossing strands in the braid corresponding to the (p, q) torus link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It then runs around the braid closure back to the top, returning to the r − q + 1 position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Together with a horizontal line from the r − q + 1 position to the r + 1 position, this strand bounds a disc in S3, lying under the plane of projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Use this disc to push the strand in S3 to become a horizontal strand lying below the plane of projection, running from the r + 1 position, then behind q strands, to the r + 1 − q position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Adjust slightly, pulling the right side up, so that the result is a closed braid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' see Figure 2, middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note that the resulting braid consists of only p − 1 strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This isotopy generalises the isotopy given by Lee in [17, Figure 6], and by de Paiva in [5, Figure 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 6 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL This move can be repeated for all the p − r strands to the right of the (r+1)-st strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When finished, we obtain a link on r strands as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Braid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Recall that the braid index of a knot K, which we will denote β(K), is the minimal number of strands required to form a braid with closure isotopic to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We will repeatedly use the following result of Franks and Williams [8] on braid index of the closure of a positive braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4 (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let B be a positive braid on p strands that contains a full twist ∆2 = (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then B has braid index p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q, d and r be positive integers such that q ≤ r < p and d + q ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let Br be a positive braid on r strands, and let Bp denote the braid on p strands obtained by adding p − r trivial strands to the right of the braid Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the closure of the braid on p strands Bp(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)d(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q has braid index equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, the closure of the given braid on p strands is equivalent to the closure of the braid on r strands B′ = (σr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−q+1)p−rBr(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)d(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σr−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because this is a positive braid, and because d + q ≥ r, the braid B′ has at least one positive full twist on r strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4 implies that the closure of B (and B′) has braid index equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose 0 < r1 < · · · < rn < p are integers, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn and q are positive integers, and suppose q ≤ rn ≤ sn + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the T-link K = T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rn, sn), (p, q)) has braid index equal to rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let Brn be the braid on rn strands obtained as the concatenation of torus braids (r1, s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (rn−1, sn−1), where we view each (ri, si) as a braid on rn strands by adding rn − ri trivial strands to the right of the braid (ri, si) = (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σri−1)si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the given T-link is the closure of the braid Brn(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrn−1)sn(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σp−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since q ≤ rn ≤ sn + q, the result follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ The next definition is from Williams [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A generalized q-cabling of a link L is a link L′ contained in the interior of a tubular neighbourhood L × D2 of L such that (1) each fiber D2 intersects L′ transversely in q points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' and (2) all strands of L′ are oriented in the same direction as L itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 7 Williams showed the following result on generalised q-cablings for knotted L in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 (Theorem 1 of Williams [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The braid index is multiplicative under generalized cabling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' That is, if L is a link with each component a non-trivial knot and L′ is a generalized q-cabling of L then β(L′) = qβ(L), where β(∗) is the braid index of ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ This result was extended to unknotted L in the case of positive braids by de Paiva in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The following result is from that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 of [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let L′ be a generalized q-cabling of the unknot L, with L given by a positive braid on n strands, where n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Also, assume the knot inside L is given by a positive braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then L′ has braid index equal to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Parents of T-links In this section, we build the “parent links” mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Dehn filling on such links produces T-links with full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By classifying when such links are hyperbolic, and applying Thurston’s hyperbolic Dehn filling theorem, we show that, in an appropriate sense, most T-links with only full twists are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is an extension of work by de Paiva and Purcell [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There, the same links were constructed, and some conditions were given to guarantee hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Here, we strengthen the result by completely characterising when such links are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers such that 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the (p, q)-torus braid on p strands, and its closure, the torus link T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let F denote the Heegaard torus on which T(p, q) lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a be an integer with 0 < a < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Denote by Ja an unknot lying horizontally with respect to the (p, q)-torus braid, positioned just above the crossings of the braid, bounding a disc such that the interior of that disc meets F transversely in a single arc intersecting the a leftmost strands of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' More generally, given a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an satisfying 1 < a1 < · · · < an < p, take disjoint unknots Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan as above, positioned so that the i-th is pushed vertically above the (i + 1)-th with respect to the braid, so that all are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Figure 3 shows an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , a1 be integers such that 1 < a1 < · · · < an < p, with n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Also, assume that there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In [6], it is shown that K is hyperbolic if all the ai > q are not multiples of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Here, we show only one needs not be a multiple of q for hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose S3 − N(K) admits an essential torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then T bounds a solid torus V that must contain at least one component of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 8 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Shows T(7, 2) augmented at the top right by J2, J3, and J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' First we show that we may choose V to contain T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For suppose V is disjoint from T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then it must contain at least one Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The component Jaj must have positive wrapping number in V , for otherwise T(p, q) and Jaj would have zero linking number, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because there is no essential torus in the exterior of the unknot in S3, it follows in this case that T is unknotted in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, T bounds a second solid torus V ′ containing T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus in all cases we may assume T bounds a solid torus containing T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As an ≥ q, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, the torus knot T(p, q) is isotopic to a closed braid with an strands so that under the isotopy, the largest unknot Jan becomes the braid axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because the isotopy moves only the right-most p − an strands, all unknots Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan are untouched by the isotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T is then contained in the solid torus S3 − N(Jan), and bounds a solid torus V containing T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It follows that Jan is disjoint from V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T must intersect the disc Dan bounded by Jan in a series of circles, with each circle bounding a meridian of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Each meridian of V can be isotoped to meet the same number of strands of T(p, q), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The boundary of a meridian defines an unknot in S3, and all such unknots are isotopic in S3 − N(K), where the isotopy is obtained by pushing the boundary of the meridian disc along the torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because T(p, q) forms a braid, it meets these discs monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let b denote the number of times that a meridian of V intersects the strands of T(p, q) on the disc Dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note b > 1, or else T would be boundary parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note also that V winds some number of times around the solid torus S3 − N(Jan), and note that each meridian of this solid torus meets exactly an strands of T(p, q), since this is the number of strands in the closed braid isotopic to T(p, q) obtained from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since V meets each LORENZ LINKS OBTAINED BY TWISTING 9 meridian of S3 − N(Jan) a total of an times, and each meridian of V meets T(p, q) a total of b times, b must divide an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' It follows that T(p, q) is a generalised b-cabling of L, where L is the core of the solid torus V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Observe that T is embedded in exterior of the torus knot S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By work of Tsau [21], there are no essential tori in a torus knot exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because b > 1, it follows that T must be compressible to its outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' That is, V is unknotted in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 implies that T(p, q) has braid index equal to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' On the other hand, the torus knot T(p, q) with 1 < q < p has braid index equal to q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' for example this follows from Franks and Williams’ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, b = q, and b divides an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hence, q divides an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By hypothesis, there is ai ∈ {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an} which is greater than q and not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since ai > q, it must be the case that Jai is disjoint from the solid torus V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since T(p, q) intersects the disc Dai bounded by Jai a total of ai times, and T(p, q) is a generalised q-cabling of L, it must be the case that L intersects the disc ai/q times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, q does not divide ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , a1 be integers such that 1 < a1 < · · · < an < p with n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 has no annuli with boundaries in two different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that S3 − N(K) has an annulus A with boundaries ∂1A and ∂2A that lie in two different components, C1 and C2, respectively, of ∂(S3 − N(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case 1: Consider first that C1 and C2 are Jaj and Jak, respectively, for some j ̸= k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note ∂1A and ∂2A are isotopic in S3−N(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The linking number between Cj and ∂jA is zero if and only if ∂jA is the longitude of Cj, in which case Cj and ∂jA are isotopic, for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose ∂1A is the longitude of C1, but ∂2A is not the longitude of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since ∂1A and ∂2A are isotopic, C1 and C2 would have nonzero linking number in this case, but this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Similarly ∂2A cannot be the longitude of C2 if ∂1A is not the longitude of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus either ∂1A is the longitude of C1 and ∂2A is the longitude of C2, or neither is a longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If both are longitudes, then C1 and C2 are isotopic, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus neither are longitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the linking number between C2 and ∂2A is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, C1 and C2 have zero linking number, so ∂1A and C2 must have zero linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But ∂2A is isotopic to ∂1A, and so ∂1A and C2 have nonzero linking number equal to the linking number of C2 and ∂2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case 2: Now suppose that C1 and C2 are Jaj and T(p, q), respectively, for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Again ∂1A and ∂2A are isotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 10 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Suppose first that ∂2A wraps at least one time along the longitude of C2 = T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then ∂2A has positive linking number with each of the components Jak, because T(p, q) has positive linking number with each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But the linking number between ∂2A and Jak for Jak ̸= C1 is zero, because C1 has linking number zero with each such component, and ∂2A has the same linking number with C1 as ∂1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus ∂2A is a meridian of C2 = T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So ∂2A and T(p, q) have linking number equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The curve ∂1A is some torus knot T(a, b) on ∂N(C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If a is equal to zero, then ∂1A is a meridian of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because a meridian of C1 has linking number zero with C2 = T(p, q), it follows that ∂1A and T(p, q) have linking number equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, this is not possible as ∂1A and ∂2A are isotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So, a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The linking number between ∂1A and C2 = T(p, q) is equal to a · aj, where C1 = Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because ∂2A and T(p, q) have linking number 1, and ∂1A and T(p, q) have linking number identical to ∂2A and T(p, q), it follows that a · aj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is impossible since aj > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, no such annulus exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let K be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K has no essential annuli with both boundary components in ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that S3 −N(K) has an essential annulus A with both bound- ary components in ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The exterior of a torus knot has just one essential annulus by work of Tsau [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By work of Lee, [16, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1] that essential annulus would be punctured by Jai, where ai > q is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A is not essential in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A is compressible, boundary compressible, or boundary parallel in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Observe that a boundary compressible annulus is in fact boundary parallel, using the fact that S3 − N(T(p, q)) is irreducible and boundary irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider first that A is boundary parallel to an annulus B in ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then A ∪ B bounds a solid torus V in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since A is not boundary parallel in S3 − N(K), at least one Jaj must be inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In addition, Jaj has wrapping number greater than zero in V , or else T(p, q) and Jaj would have linking number equal to zero, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But Jaj is an unknot, whose complement admits no essential tori (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [13, page 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus V is also unknotted in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This implies that B is a meridional annulus of ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If ∂V is boundary parallel to Jaj, then Jaj is the core of ∂V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hence, the linking number between T(p, q) and Jaj would be one, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, as ∂V is not boundary parallel to Jaj, ∂V is an essential torus for S3 − N(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Assume now that A is compressible in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then there is a compression disk D for A in S3 − N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Surgering A along D yields two discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since S3 − N(T(p, q)) is boundary irreducible, ∂Di bounds a disk Ei on ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, by pushing Ei slightly off of ∂N(T(p, q)) in S3 −N(K), we obtain a compressing LORENZ LINKS OBTAINED BY TWISTING 11 disc for A in S3−N(K), which contradicts our assumption that A is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, A is not compressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A cannot have both boundary components on ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let K be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K has no essential annulus with both boundary components on one ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that S3 −N(K) has an essential annulus A with both bound- ary components on ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since S3 −N(Jaj) is a solid torus, and the solid torus admits no essential annuli, A is not essential in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus A is either compressible or boundary parallel in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case A: Suppose A is boundary parallel, parallel to an annulus B in ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then A ∪ B bounds a solid torus V in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since A is not boundary parallel in S3 − N(K), at least one component C of K must be inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case A1: Consider first that C = T(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then T(p, q) has wrapping number greater than zero in V , for otherwise Jaj and T(p, q) would have zero linking number, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note this implies that ∂V is incompressible to its inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that some circle Jak with j ̸= k lies in S3 − V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then we may isotope Jak to lie outside of W = N(Jaj) ∪ V , which is a regular solid torus neighbourhood of the unknot Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Denote by ω the winding number of Jak in S3 − W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If ω = 0, then the linking number between Jak and T(p, q) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus, ω ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But then this implies that the linking number between Jaj and Jak is nonzero, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus all circles Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jai−1, Jai+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan are inside V in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because at least two components of K lie inside V , ∂V is not boundary parallel to the inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The core of V forms a torus knot T(a, b) on N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Note b > 0 or else T(p, q) runs around a longitude of N(Jaj) and hence has linking number zero with Jaj, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose b = ±1, so the core of V has the form of the trivial knot T(a, ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then there exists a disc in S3 − N(K) that is a longitude for ∂N(Jaj) whose boundary can be divided into two arcs, one of which meets A ⊂ ∂V in a nontrivial arc, and the other meets ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is an essential boundary compression disc for A, contradicting the fact that A is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since |b| > 1, ∂V = ∂N(T(a, b)) is incompressible and not boundary parallel to the outside, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' in the solid torus S3 − Jaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This implies that in all cases ∂V is essential in S3 − N(K) contradicting Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case A2: The torus knot T(p, q) cannot lie inside V by the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So some C = Jak with j ̸= k lies inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The wrapping number of Jak inside V must be different from zero as Jak and T(p, q) have positive linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since Jak and Jaj have zero linking number, V must be a longitude of ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If Jak is the core of V , then Jaj and Jak are isotopic 12 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A disc with boundary an arc on each of A ⊂ ∂V and ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' in S3 − N(T(p, q)), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So Jak is not the core of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But then ∂V is incompressible and not boundary parallel to the inside in S3 − K, and incompressible and not boundary parallel to the outside in S3 − K, contradicting Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Case B: Suppose A is compressible in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then there is a compression disk D for A in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Surgering A along D yields two discs, D1 and D2, such that ∂A = ∂D1 ∪ ∂D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If one of ∂D1 or ∂D2 bounds a disk E on ∂N(Jaj), then by considering a disc with boundary in A close to E, we see that A is also compressible in S3 − N(K), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So suppose that neither ∂D1 nor ∂D2 bounds a disk on ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then D1 and D2 are discs in the solid torus S3 − N(Jaj) with nontrivial boundary on ∂N(Jaj) and hence both are meridians of S3 − N(Jaj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' with ∂D1 and ∂D2 forming longitudes of ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Undoing the surgery along D, it follows that A is boundary parallel in S3 − N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus we have a contradiction to Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, S3 − N(K) has no essential annulus with both boundary com- ponents in one ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The link K as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 has no essential annuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, any essential annulus has both boundary components on the same component of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4, the two boundary components cannot lie on ∂N(T(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5 the two boundary components cannot lie on one of the ∂N(Jaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus no such annulus exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , a1 be integers such that 1 < a1 < · · · < an < p with n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Also, assume that there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, the link K = T(p, q) ∪ Jan ∪ · · · ∪ Ja1 is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By de Paiva and Purcell [6, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1], the link exterior is irre- ducible and boundary irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, it is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6, it is anannular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore it is hyperbolic by Thurston’s hyperbolisation theorem for Haken manifolds [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7 and de Paiva and Purcell [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6], we completely classify the geometric types of the links T(p, q) ∪ Ja1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an be integers such that 1 < a1 < · · · < an < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the link K = T(p, q) ∪ Ja1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Jan is hyperbolic if and only if either all ai < q, or there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When n = 1, the link K = T(p, q) ∪ Ja1 is the Dehn-filling parent of a twisted torus knot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' this has been treated by Lee [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If n = 1 and a1 = q, then [15, Theorem 1] implies that infinitely many Dehn surgeries along Ja1 yield non-hyperbolic knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, Thurston’s hyperbolic Dehn filling theorem [19] implies K is not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In fact, the proof of [15, Theorem 1] implies K is annular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If n = 1 and a1 is not a multiple of q, then K is hyperbolic by [16, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In the case n > 1, if there is ai > q that is not a multiple of q, then K is hyperbolic by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If n > 1 and all ai are less than q, then no ai is a multiple of q, and K is hyperbolic by [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, if n > 1, there is some ai > q and all ai > q are multiples of q, then K is satellite by [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be relatively prime integers with 1 < q < p, and let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn be integers such that 1 < a1 < · · · < an < p and si > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then, there exists B ≫ 0 such that if each si > B, the T-link T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) is hyperbolic if and only if either all ai < q, or there is ai > q which is not a multiple of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8, the link K = T(p, q) ∪ Ja1 ∪ · · · ∪ Jan is hyperbolic if and only if the ai satisfy the hypotheses of the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Obtain the given T-link by Dehn filling the link components Ja1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Jan along slopes 1/s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , 1/sn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When the link K is hyperbolic, the Dehn filling remains hyperbolic by Thurston’s hyperbolic Dehn filling theorem [19] pro- vided the si are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' On the other hand, Dehn filling a satellite K yields a satellite T-link, by de Paiva and Purcell [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6], and in the case n = 1 and a1 = q, Dehn filling yields an annular link by Lee [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1 in the introduction follows immediately from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hyperbolicity with effective full twist bounds While Corollary Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 is quite broad, unfortunately the constant B in that theorem is not explicit, and so it may be difficult to apply in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In this section we find explicit parameters which produce hyperbolic T-knot obtained by full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because we are considering full twists exclusively in this section, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 implies that we may assume that none of the ai are equal to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 14 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an, and 0 < q < an < p, each si > 0, and sn ≥ 2, p and an are relatively prime, k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the T-knot K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q + kp)) is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose that the exterior of K in S3 admits an essential torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By work of Ito [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2(3)], because K is the closure of a braid with at least two positive full twists on p strands, the torus T does not intersect the braid axis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Moreover, the knot inside T is given by a braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus there exists some integer d > 1 such that K is a generalized d-cabling of a knot L, where L is the core of the solid torus bounded by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As a consequence, d must divide p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' After (−1/k)-Dehn surgery along the braid axis C, the knot K becomes the T-knot K′ = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) and the torus T becomes a new torus T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This will bound a solid torus V ′ in S3, with core L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because q < an < ansn + q, the knot K′ has braid index equal to an by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If L′ is trivial, then an is equal to d by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' However, this is not possible since gcd(p, an) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So L′ is knotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8, an is equal to dβ(L′), where β(L′) is the braid index of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' But then d divides p and d divides an, again contradicting gcd(p, an) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, the exterior of K admits no essential torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ We will combine the previous result with the following from [5], which gives information on torus knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2 of de Paiva [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn be positive integers such that 1 < q < p and 1 < a1 < · · · < an < p with ai ̸= q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If gcd(p, q) = 1 and in addition one of the following hold: q < an, or q > an and p is not of the form bq + 1 for some b > 0, or q > an and p = bq + 1 for some b > 0, but s1 > 1, or q > an, p = bq + 1 for some b > 0, and s1 = 1, but a2 ̸= a1 + 1, then T((a1, s1a1), (a2, s2a2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, snan), (p, q)) is not a torus knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an and 1 < q < an < p, each si > 0 and sn ≥ 2, LORENZ LINKS OBTAINED BY TWISTING 15 p and an are relatively prime, k ≥ 2, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then if in addition, one of the following hold: q ̸= 1, or s1 > 1, or a2 ̸= a1 + 1, then the T-link K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q + kp)) is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because gcd(p, q) = 1, K is a knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1, K is atoroidal, so not a satellite knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The T-knot K is equivalent to the T-knot T((a1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (q + kp, p)) by [1, Corollary 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The integer q + kp does not have the form bp + 1 if and only if q is different from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So under these conditions, K is not a torus knot by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, by Thurston’s hyperbolisation Theorem for knots [20], K is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an, and 1 < q < an < p, each si > 0 and both sn and sn−1 are at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose also that one of the following holds: q < an−1 and an and an−1 are relatively prime, or q > an−1 and an and q are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the knot K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose the exterior of K in S3 admits an essential torus T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, K is equivalent to the knot given by the closure of the braid B = (σan−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−q+1)p−an · τ · (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−1)snan+q, where τ is the concatination of braids (a1, a1s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an−1, an−1sn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since B has at least two positive full twists on an strands, it follows from [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2(3)] that T does not intersect the braid axis C of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus there is an integer d > 0 such that K is a generalized d-cabling of the core L of the solid torus bounded by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hence d divides an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Perform (−1/sn)-Dehn surgery along the braid axis C to obtain the braid B′ = (σan−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−q+1)p−an · τ · (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σan−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Its closure gives K′ = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an−1, an−1sn−1), (an, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T becomes a new essential torus T ′ in the exterior of K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus T ′ bounds a solid torus with core L′, which is either trivial or knotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 16 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL Suppose first the case that q < an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then q ≤ an−1 ≤ an−1sn−1 + q, so Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='6 implies that K′ has braid index equal to an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If L′ is the trivial knot, then an−1 is equal to d by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This implies that d divides both an and an−1, contradicting the assumption in this case that these are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Similarly, if L′ is knotted, then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 implies that an−1 is a multiple of d, with the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Now suppose q > an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K′ has braid index q by Franks and Williams, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' If L′ is trivial, then as above, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='9 implies q equals d, and therefore d divides both an and q, contradicting the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Similarly, if L′ is knotted, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='8 implies q is a multiple of d, and again d divides both an and q, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , an, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sn, and p, q, k be integers satisfying the following hypotheses: p and q are relatively prime, 1 < a1 < · · · < an and 1 < q < an < p, each si > 0 and both sn and sn−1 are at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose also that one of the following holds: q < an−1 and an and an−1 are relatively prime, or q > an−1 and an and q are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then K = T((a1, a1s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, ansn), (p, q)) is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4, the knot K is atoroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, using the fact that q < an, K is anannular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Therefore, K is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Satellite T-links obtained by Half-twists In this section we switch from discussions of hyperbolic links to satellite links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We find families of Lorenz links that are satellites using half-twists, rather than full-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Previous work by de Paiva and Purcell found con- ditions that ensure a T-link is satellite, namely [6, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lee has similar results for the case of twisted torus knots [15, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' We extend these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Suppose B is a diagram given as a closed braid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' we consider the braid to have strands running vertically on the plane of projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A positive half-twist on the strands from a to b is the braid ∆a,b = (σa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σb)(σa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σb−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This can be thought of as cutting the braid between the a-th and b-th strands, rotating in the anticlockwise direction by 180◦, and gluing back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In braid theory literature, the positive half-twist on all strands is well known as the Garside fundamental braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A negative half-twist is defined similarly, only the rotation is in the clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' See Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 17 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' An example of half twists when r = 2, q = 3, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Left: A positive half-twist ∆1,rq, a negative half-twist ∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The green circle indicates the braid axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Middle: The negative half- twist cancels crossings above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Right: The additional positive half-twist gives the braid (rq, tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let r, q, s be positive integers, and suppose s is not a multiple of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The (rq, sq)-torus braid is obtained by the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Start with the trivial braid on rq strands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' let J1,rq be an unknot encircling all rq strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let t be an integer such that 0 < t < r and s = t + kr for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Insert a positive half-twist ∆1,rq, followed by a negative half-twist ∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, perform 1/k-Dehn filling on J1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The result is the (rq, sq)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The process is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The positive half-twist ∆1,rq yields a braid (σ1σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrq−1)(σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrq−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σ1), encircled by J1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Perform the negative half-twist ∆1,(r−t)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This concate- nates the previous braid with (σ−1 (r−t)q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ−1 2 σ−1 1 )(σ−1 (r−t)q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ−1 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (σ−1 (r−t)q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This braid cancels with the positive half-twist along the first (r − t)q strands, as shown in Figure 5, middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, the positive half-twist ∆(r−t)q+1,rq concatenates a positive half-twist along the last tq strands, giving the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrq−1)tq = (rq, tq), still augmented by the unlink J1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' To obtain the braid (rq, sq), perform 1/k Dehn filling on J1,rq, removing that link component and inserting an additional krq overstrands into the braid, for a total of tq+krq = sq overstrands, giving the desired (rq, sq)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let r, q, s be positive integers, with s not a multiple of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Consider the torus braid (rq, sq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' At the top of the braid, consider r disjoint discs arranged horizontally, each encircling q strands of the braid, and similar discs at the bottom of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The boundary of each disc at the top connects 18 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL via a cylinder, embedded in the complement of the braid and enclosing q strands, to the boundary of a disc at the bottom of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Moreover, the solid cylinders enclosed by these cylinders, containing q strands each, forms the (r, s)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let t be an integer such that 0 < t < r and s = t+kr for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, the (rq, sq) torus braid is formed from k full twists on rq strands, followed by a positive half-twist ∆1,rq, then a negative half-twist −∆1,(r−t)q and a positive half-twist ∆(r−t)q+1,rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Each half-twist is on a multiple of q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Observe that the cylinders described above can be arranged to completely contain any half-twist on q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' For a half-twist on a multiple of q strands, say xq strands, x disjoint cylinders enter the top of the half-twist, and then are half-twisted themselves, remaining disjoint, to exit the bottom of the half-twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus the cylinders remain embedded as claimed when passing through half-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, each full twist also preserves the cylinders, sending each through a full twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' To see that the braid formed by the solid cylinders is as claimed, observe that the cylinders form k full twists, followed by one positive half-twist on all strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The (r−t) left-most cylinders then pass through a negative half-twist, and the remaining t right-most cylinders pass through a positive half-twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='2, this creates braid on r strands, with rk overstrands at the top coming from the full twists, followed by t overstrands coming from the concatenation of half-twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus this is an (r, rk + t) = (r, s)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Let p, q be integers such that 1 < q < p, and let (a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn) be pairs of integers such that 1 < a1 < · · · < an ≤ q and bi > 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , rm and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , sm be integers such that q < r1q < · · · < rmq < p, and si > 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then the T-link K = T((a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn), (r1q, s1q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rmq, smq), (p, q)) is satellite with companion the T-link T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rm, sm+1)) and pattern given by the closure of the braid (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn)(σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm � q, q � m � i=1 risi � + qrm � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As before, we think of the T-link as the closure of a braid on p strands arranged vertically, the concatenation of braids (a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn), (r1q, s1q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rmq, smq), (p, q) in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' First apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3 to change the closed braid of the T-link to a closed braid B′ on rmq strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This isotopy fixes all of the rmq strands at the top left of the original braid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' thus it does not affect any of the braids (aj, bj) or (riq, siq), for any i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In other words, B′ is the braid given by concatenating (σrmq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−q+1)p−rm with braids (a1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (an, bn), (r1q, s1q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , (rmq, smq), and finally the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' LORENZ LINKS OBTAINED BY TWISTING 19 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, there are rm disjoint embedded cylinders in the complement of the portion of the braid starting just above the braid (a1, b1), and ending just below the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−1)q at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These cylinders each enclose q strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' They extend around the braid closure to give rm disjoint embedded cylinders running to the top of the braid, each enclosing q strands, arranged right to left across the top of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The only portion of the braid that is not already enclosed in one of these cylinders is the braid (σrmq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−q+1)p−rm lying at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is a braid whose left-most strand is the (rmq − q + 1)-th strand, and whose right-most strand is the rmq-th strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' In other words, this is a braid on the right-most q strands of the rmq-strand braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus the right-most cylinder, enclosing q strands, can be extended to enclose this braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Then all cylinders connect to form a closed embedded torus Σ, encircling q strands of the braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The torus Σ bounds a solid torus containing q strands, which we check has the claimed form of the companion in the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This solid torus forms a braid on rm strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='3, each (riq, siq)-torus braid from the original T-link causes the solid cylinder to form a braid (ri, si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The braids (aj, bj) and (σrmq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−q+1)p−rm lie completely inside the solid cylinder, so they do not affect the braid it forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally, consider the braid (σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σrmq−1)q at the bottom of B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is formed by q strands running over all the rmq strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' When the collection of solid cylinders encounter this braid, the left-most solid cylinder encircles exactly these q strands, and runs over all others to lie on the right-most side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus it forms a (rm, 1)-torus braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' So the solid torus enclosing q strands has the form of the closure of a braid (r1, s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (rm, sm), (rm, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' This is the T-link T((r1, s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (rm, sm + 1)) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Since it forms a nontrivial knot in S3, Σ is an incompressible torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Finally we check the form of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Starting at the top-left of the braid B′, the torus Σ encloses the braid (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn), which will form part of the braid describing the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' As Σ follows the companion into each of the braids (ri, si), all the q strands will make one full twist each time Σ runs completely through an overstrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' There are si of these, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' m − 1, plus sm + 1 for the (rm, sm + 1) braid that the compan- ion runs over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' These will occur in some order, with Σ also enclosing the braid (σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm, coming from the top right of B′, at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Because full twists commute in the braid group, we may write the braid as (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn)(σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm · τ where τ is an appropriate number of full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' To obtain the appropriate number of full twists, we need to consider the homological longitude of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The pattern is the braid obtained when we apply a homeomorphism taking the solid torus bounded by the companion to an unknotted solid torus, with homological longitude mapped to a standard longitude of the unknot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' The effect is to add �m−1 i=1 (ri − 1)si + (rm − 1)(sm + 1) additional full twists, for a total of 20 THIAGO DE PAIVA AND JESSICA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' PURCELL �m i=1 risi + rm full twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thus the pattern can be written as the braid (a1, b1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (an, bn)(σq−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' σ1)p−rm(q, q( � risi) + qrm) □ References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Joan Birman and Ilya Kofman, A new twist on Lorenz links, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2, 227–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 2529294 [1, 4, 15] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Joan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Birman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Williams, Knotted periodic orbits in dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lorenz’s equations, Topology 22 (1983), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 47–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 682059 [1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thiago de Paiva, Hyperbolic knots given by positive braids with at least two full twists, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 150 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 12, 5449–5458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 4494619 [7] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Satellite knots over lorenz knots which are not lorenz knots, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='12816, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [1] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Torus Lorenz links obtained by full twists along torus links, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=', to appear (2022), arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='10935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [2, 5, 14] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thiago de Paiva and Jessica S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Purcell, Satellites and Lorenz knots, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=', to appear (2021), arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='09500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [1, 2, 3, 4, 7, 12, 13, 16] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' El-Rifai, Necessary and sufficient condition for Lorenz knots to be closed under satellite construction, Chaos Solitons Fractals 10 (1999), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 137–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 1682295 [1] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' John Franks and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Williams, Braids and the Jones polynomial, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 303 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 97–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 896009 [6] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Ghys and J Leys, Lorenz and modular flows: a visual introduction, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='ams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='org/publicourtreach/feature-column/fcarc-lorenz, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [1] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Paulo Gomes, Nuno Franco, and Lu´ıs Silva, Partial classification of Lorenz knots: syllable permutations of torus knots words, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' D 306 (2015), 16–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 3367570 [1] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Farey neighbors and hyperbolic Lorenz knots, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Knot Theory Ramifications 26 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 9, 1743004, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 3687479 [1] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' John Guckenheimer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Williams, Structural stability of Lorenz attractors, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (1979), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 50, 59–72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 556582 [1] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Allen Hatcher, Notes on basic 3-manifold topology, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [10] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Tetsuya Ito, Braid ordering and the geometry of closed braid, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 15 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 473–498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 2788641 [14, 15] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Sangyop Lee, Twisted torus knots T(p, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' kq, s) are cable knots, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Knot Theory Rami- fications 21 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 1250005, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 2887898 [13, 16] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Twisted torus knots that are unknotted, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' IMRN (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 18, 4958–4996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 3264672 [2, 10, 13] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Positively twisted torus knots which are torus knots, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Knot Theory Ramifi- cations 28 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 3, 1950023, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 3938086 [5] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Lorenz, Deterministic non-periodic flow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 20 (1963), 130–141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [1] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Thurston, The geometry and topology of three- manifolds, Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Notes, 1979, Available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='msri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='org/communications/books/gt3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [13] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' , Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=') 6 (1982), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 3, 357–381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' [1, 12, 15] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Chichen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Tsau, Incompressible surfaces in the knot manifolds of torus knots, Topology 33 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 197–201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 1259522 [9, 10] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 1, 53–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 1870856 [1] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Williams, The braid index of generalized cables, Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 155 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' 2, 369–375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'} +page_content=' MR 1178031 [6, 7]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQf-f5K/content/2301.01934v1.pdf'}