The Michigan Mathematical Journal
Advance publication
Michigan Math. J.
Volume 67, Issue 3 (2018), 647-672.
Extensions of Some Classical Local Moves on Knot Diagrams
Benjamin Audoux, Paolo Bellingeri, Jean-Baptiste Meilhan, and Emmanuel Wagner
More by Benjamin Audoux
More by Paolo Bellingeri
More by Jean-Baptiste Meilhan
More by Emmanuel Wagner
We consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass, and welded band-pass moves. Interrelationships between these moves are discussed, and, for each of these moves, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a byproduct, we obtain that all of the local moves mentioned are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.
Michigan Math. J., Volume 67, Issue 3 (2018), 647-672.
Received: 16 December 2016
Revised: 20 June 2017
First available in Project Euclid: 13 July 2018
https://projecteuclid.org/euclid.mmj/1531447373
doi:10.1307/mmj/1531447373
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 20F36: Braid groups; Artin groups
Audoux, Benjamin; Bellingeri, Paolo; Meilhan, Jean-Baptiste; Wagner, Emmanuel. Extensions of Some Classical Local Moves on Knot Diagrams. Michigan Math. J. 67 (2018), no. 3, 647--672. doi:10.1307/mmj/1531447373. https://projecteuclid.org/euclid.mmj/1531447373
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University of Michigan, Department of Mathematics
Hochster Volume 57
On usual, virtual and welded knotted objects up to homotopy
AUDOUX, Benjamin, BELLINGERI, Paolo, MEILHAN, Jean-Baptiste, and WAGNER, Emmanuel, Journal of the Mathematical Society of Japan, 2017
Arrow calculus for welded and classical links
Meilhan, Jean-Baptiste and Yasuhara, Akira, Algebraic & Geometric Topology, 2019
Crossing changes, Delta moves and sharp moves on welded knots
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Isotopy and homotopy invariants of classical and virtual pseudoknots
Dorais, François, Henrich, Allison, Jablan, Slavik, and Johnson, Inga, Osaka Journal of Mathematics, 2015
Braid presentation of virtual knots and welded knots
Kamada, Seiichi, Osaka Journal of Mathematics, 2007
$\mathit {SH}(3)$-move and other local moves on knots
Kanenobu, Taizo, Osaka Journal of Mathematics, 2014
Region crossing change is an unknotting operation
SHIMIZU, Ayaka, Journal of the Mathematical Society of Japan, 2014
Minimal surface representations of virtual knots and links
Dye, H A and Kauffman, Louis H, Algebraic & Geometric Topology, 2005
The Unknotting number and band-unknotting number of a knot
Abe, Tetsuya, Hanaki, Ryo, and Higa, Ryuji, Osaka Journal of Mathematics, 2012
An Unknotting Index for Virtual Knots
KAMADA, Seiichi, KAUR, Kirandeep, KAWAUCHI, Akio, and MADETI, Prabhakar, Tokyo Journal of Mathematics, 2019
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