Patterns of the $V_2$-polynomial of knots
Stavros Garoufalidis Shana Yunsheng Li







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    Patterns of the V_2-polynomial of knots]
    Patterns of the V_2-polynomial of knots
             
      International Center for Mathematics, Department of Mathematics 
    
      Southern University of Science and Technology 
    
      Shenzhen, China <http://people.mpim-bonn.mpg.de/stavros>stavros@mpim-bonn.mpg.de
      Department of Mathematics 
    
      University of Illinois 
    
      Urbana, IL, USA <https://li-yunsheng.github.io>yl202@illinois.edu Key words and phrases:
      Knots, Jones polynomial, V_n-polynomial, Nichols algebras, R-matrices,
      Yang–Baxter equation, knot polynomials, knot genus, Conway mutations, Khovanov
      Homology, Heegaard Floer Homology, homologically thin and thick knots, tight and
      loose knots.
    
    § ABSTRACT 
      Recently, Kashaev and the first author defined a sequence V_n of 2-variable
      knot polynomials with integer coefficients, coming from the R-matrix of a rank
      2 Nichols algebra, the first polynomial been identified with the Links–Gould
      polynomial. In this note we present the results of the computation of the
      V_n polynomials for n=1,2,3,4 and discover applications and emerging patterns,
      including unexpected Conway mutations that seem undetected by the V_n-polynomials
      as well as by Heegaard Floer Homology and Knot Floer Homology. 
    [
        Shana Yunsheng Li
        5 September 2024
    =====================
    
    
    
    
    § INTRODUCTION
    
    
     §.§ A sequence of 2-variable knot polynomials
    
    
    Recently, Rinat Kashaev and the first author defined multivariable polynomials
    of knots using Nichols algebras <cit.> with automorphisms. In our paper,
    we focus on the sequence of 2-variable polynomial invariants V_n(t,q) of knots
    which come by applying a general construction of <cit.> to one of the
    simplest Nichols f-algebras of rank 2 of diagonal type.
    This algebra depends on one variable q that determines the braiding and two
    variables (t_1,t_2) that determine the automorphism type, and when q is not
    a root of unity and (t_1,t_2,q) satisfy the relation t_1t_2q^n=1 for some
    positive integer n>0, then the Nichols algebra has a right Yetter–Drinfel'd
    f-module Y_n of dimension 4n <cit.> and an explicit
    R-matrix T_n. Taking this as a black box, and parametrizing the three variables
    (t_2,t_2,q) satisfying t_1t_2q^n=1 in terms of two variables (t,q)(t_1,t_2)=(1/(q^n/2 t,t/q^n/2), it was shown in <cit.> that Y_n
    comes equipped with an R-matrix T_n which leads to a matrix-valued knot invariant
    K ↦ J_T_n(K) ∈End(Y_n)
    as well as to a scalar-valued invariant
    V_K,n(t,q)∈[t^± 1,q^± 1/2] given by the (1,1)-entry of
    J_T_n(K).
    
    It was advocated in <cit.> that the sequence V_n of 2-variable knot
    polynomials has similarities and differences with the sequence of the Jones
    polynomial of a knot and its parallels, otherwise known as the colored Jones
    polynomial.
    
    The polynomial invariant V_n satisfies
    
      * 
      the symmetry
    
    V_K,n(t,q)=V_K,n(t^-1,q),      V_K,n(t,q)=V_K,n(t,q^-1) 
    
    where the first equality comes from the involution exchanging t_1 and t_2,
    and in the second one K denotes the mirror image of K, 
    
      * 
      conjecturally, the specialization
      
    V_K,n(q,q)=1,      V_K,n(t,1) =Δ_K(t)^2
    
    where Δ_K(t) ∈[t^± 1/2] is the symmetrized
    (i.e., Δ_K(t)=Δ_K(t^-1)), normalized (i.e., Δ_K(1)=1)
    Alexander polynomial,
    
      * 
      conjecturally, the relation V_1= with the Links–Gould
      invariant <cit.>,
    
      * 
      and conjecturally, the genus bound
    _t V_K,n≤ 4 g(K)
    
    where the Seifert genus g(K) is the smallest genus of a spanning surface of a knot.
    Here, by t-degree of a Laurent polynomial of t we mean the difference between
    the highest and the lowest power of t.
    
    
    The paper <cit.> stimulated a lot of subsequent work. 
    The relation V_1= is now known <cit.>, and consequently
    the specialization (<ref>) and the genus bounds (<ref>) holds for
    n=1 since they hold for the Links–Gould invariant <cit.>.
    On the other hand, it is known that the Links–Gould invariant does not detect
    Conway mutation, whereas the genus does, and hence the genus bounds (<ref>)
    cannot be sharp for n=1.
    
    More recently, an expression of V_2 in terms of V_1 polynomial of a knot
    and its (2,1)-parallel was obtained <cit.>, and this proves both the
    specialization (<ref>) and the genus bound (<ref>) for n=2. 
    
    But of all those properties of V_2, the last inequality is the most intriguing.
    Based on some experiments with few knots and 12 and 13 crossings, it was observed
    in <cit.> that in all computed cases, the inequality (<ref>) is
    in fact an equality. Is this an accident for knots with low numbers of
    crossings, or a new phenomenon? To decide one way or another, one needs an efficient
    way to compute the V_n-polynomials of knots, do so, and sieve the data.
    This is exactly what we did, and led to the results of our paper.
    
    
    
     §.§ Is the genus bound an equality for V_2?
    
    
    Since we are talking about tables of knots and their invariants, we will be 
    using the naming of the HTW table of knots up to 16 crossings <cit.>
    imported in  <cit.> and also in
     <cit.>.
    
    
    
    The inequality (<ref>) combined with the specialization (<ref>)
    for n=2 imply that
    
    2 _t A_K(t) ≤_t V_K,2(t,q) ≤ 4 g(K)  .
    
    On the other hand,
    _t A_K(t) ≤ 2 g(K)
    
    with equality if and only if a knot is Alexander-tight, otherwise Alexander-loose.
    In our paper we abbreviate these two classes simply with tight/loose, similar to what
    people do in Heegaard Floer Homology and Khovanov Homology where they talk about
    HFK-thin/thick or Kh-thin/thick knots, but then they drop the HFK or Kh once
    the context is clear. Likewise, we use the term thin in our paper to mean HFK and
    Kh-thin. Note that alternating knots are tight <cit.>.
    What's more, quasi-alternating knots (a class that includes all alternating knots)
    introduced by Ozsvath-Szabo in <cit.> are HFK and
    Khovanov-thin <cit.>, and hence tight.
    
    Combining the above two inequalities, it follows that
    the inequality (<ref>) is in fact an equality for V_2 and all tight knots.
    Note next that there are no loose knots with ≤ 10 crossings. Moreover, the number
    of loose knots with ≤ 16 crossings is given in Table (<ref>).
    Incidentally, the list of loose knots was compiled by computing in 
    the Alexander polynomial, and also the HFK (and in particular, the Seifert genus
    of a knot).
    
    Among the loose knots, are the ones with trivial Alexander polynomial (also computed
    by ) which are in some sense extreme. The list of 7 loose knots
    (up to mirror) with 11 crossings is
    
    11n34^∗,  11n42^∗,  11n45,  11n67,
     11n73,  11n97,  11n152 
    
    where the asterisque indicates that the knot has trivial Alexander polynomial, and
    the pair (11n34,11n42) is the famous Kinoshita–Terasaka and Conway pair of
    mutant knots. Their genus is given by 3, 2, 3, 2, 3, 2, 3, the t-degree of the
    V_1-polynomial is 6, 6, 8, 6, 8, 6, 8 and the t-degree of the V_2-polynomial
    is 12, 8, 12, 8, 12, 8, 12, confirming the equality in (<ref>) for n=2.
    
    
    
    
    Table <ref> summarizes the knots for which the V_n-polynomial was computed.
    For n=2, we computed its values for all loose knots with at most 15 crossings
    and all trivial Alexander knots with at most 16 crossings. In all cases, we found that
    the inequality (<ref>) is an equality for n=2. Combined with the
    specialization and the genus bounds for V_2, this implies the following.
    
    
    Equality holds in (<ref>) for n=2 and for all knots with at most 15 crossings
    and all trivial Alexander polynomial knots with 16 crossings.
      
    The relation between V_2 and V_1 discussed in Section <ref> below,
    combined with the fact that V_1= implies that the map
    
    K ↦ V_K,2(e^ħ + N,e^ħ) ∈[N][[ħ]]
    
    is a Vassiliev power series invariant of knots. Hence, if  (<ref>) is an
    equality for n=2, it follows that Vassiliev invariants determine the Seifert genus
    of a knot. A celebrated method to detect the genus of the knot is Heegaard Floer
    Homology <cit.>. A second (conjectural) method to compute the genus of a knot uses
    hyperbolic geometry, and more specifically the degree of the twisted torsion
    polynomial τ_K,3(t) of a hyperbolic knot (twisted with the
    adjoint representation of the geometric representation of a hyperbolic knot); see
     <cit.>. Curiously, the Conjecture 1.7
    of <cit.> was verified for all hyperbolic knots with
    at most 15 crossings. 
    
    
     §.§ V_2-trivial Conway mutations
    
    
    A question that we discuss next is how strong is the new V_2 polynomial in separating
    knots. Given the values of the polynomial for knots up to 14 crossings, we searched
    for repetitions, taking into account mirror image, which changes V_2(t,q) to
    V_2(t,q^-1). Here, we came across a new surprise. The V_2 polynomial separates
    knots with at most 11 crossings, but fails to separate 12 crossing knots, and
    the three pairs that we found are
    
    (12n364,    12n365)        
    (12n421,    12n422)        
    (12n553,    12n556)  . 
    
    We tried the V_1 polynomial on them and it failed to
    separate them, and we tried the V_3-polynomial which also failed to separate. Yet,
    the genus inequality (<ref>) was an equality for n=2, which meant that these
    3 pairs have equal genus (in each pair). We checked their HFK,
    computed by , and their Khovanov Homology, computed by
    , and a bit to our surprise, was equal in each pair.
    Looking at these 3 pairs more closely, we realized that they are in fact
    Conway mutants. A table of mutant knots with at most 15 crossings is given in
    Stoimenow <cit.>. As was pointed out to us by N. Dunfield, one
    can separate the knots in these pairs using the homology of their 5-fold covers,
    or the certified isometry signature of the complete hyperbolic structure.
    
    Having found these unexpected pairs of Conway mutant knots, we tried knots with
    13 crossings, where we now found 26 more pairs with exactly the same properties as
    above, given in Table (<ref>). We then searched knots with
    14 crossings, where we now found 192 pairs and 1 triple (up to mirror image)
    with the same properties as above. The V_2-equivalence classes of knots of size
    more than 1 with at most 12, 13 and 14 crossings are given Tables (<ref>),
     (<ref>) and (<ref>), respectively.
    
    
    
    Summarizing, we obtain the following.
    
    
    The V_2 polynomial separates knots with at most 14 crossings except for the
    pairs and triples in Tables (<ref>),  (<ref>) and
     (<ref>)–(<ref>). The knots in those tuples with at most
    14 crossings have
    
      * 
      equal V_1, V_2, V_3 and (for at most 13 crossings) V_4
      polynomials,[with 2 exceptional pairs (14n2423, 14n5868) and
      (14n5822, 14n5852)  that have different  V_3-polynomials]
      * 
      equal HFK and equal Khovanov Homology,
    
      * 
      and they are Conway mutant knots, in particular they have equal volumes, trace field,
      colored Jones polynomials, ADO polynomials and HOMFLY polynomial.
    
    
    For lack of a better name, let us say that two knots are V_2-equivalent
    if they have equal V_2-polynomial. This notion is similar to the almost-mutant
    knots of <cit.>.
    
    
    
    
    
    
    
    Are V_2-equivalent knots always Conway mutant? Do they have equal
    equal HFK and equal Khovanov Homology? And why?
    
    
    We can give a partial answer to this question as follows. The observed
    V_2-equivalence classes come in 3 flavors
    
    
    1: tight + thin,      2: tight + thick,
         3: loose + thick .
    
    
    The HFK homology of an HFK-thin (resp., Kh-thin) knot is determined by its
    Alexander polynomial (resp., by the Jones polynomial and the signature). Since
    mutation does not change the Alexander polynomial, nor the Jones polynomial, nor the
    signature, it follows that mutant thin knots have equal HFK and equal Khovanov
    Homology. This gives an explanation of the above question for the class 1. 
    
    As mentioned in the introduction, quasi-alternating knots are tight + thin.
    On the other hand, our tables (given in the Appendix) give concrete examples
    of tight + thick or loose + thick knots. Several methods of constructing knots
    with equal HFK and equal Khovanov Homology, are discussed in detail in
    Hedden–Watson, <cit.>, but we do not know how to apply these
    constructions to generate our examples. 
    
    It is also worth noticing that all tight + thin knots listed in the appendix
    are Kh'-thin, as defined in <cit.>.
    
    
    
     §.§ Independence of the V_2 and the 2-loop polynomials
    
    
    The colored Jones polynomial can be decomposed into loop invariants, starting
    from 0-loop which is the inverse Alexander polynomial, and then going up the loops.
    In fact, the 2-loop invariant of the Kontsevich integral of a knot is essentially
    2-variable polynomial invariant, and its image under the 𝔰𝔩_2()-weight
    system is a 1-variable polynomial computed efficiently by Bar-Natan and van der Veen
     <cit.>.
    
    One may ask whether the V_2-polynomial is determined by the 2-loop invariant Z_2
    of the Kontsevich integral. It is known that the map K ↦ Z_2(Wh(K))
    is a degree 2 Vassiliev invariant of knots <cit.>, where
    Wh(K) denotes the Whitehead doubling of a 0-framed knot with a positive
    clasp. Since the vector space of degree 2 Vassiliev invariants is 1-dimensional
    generated by a_2(K)=Δ”(1), it follows that
    Z_2(Wh(K)) = c   a_2(K) for a universal nonzero constant c.
    For the trefoil and its mirror image, we have a_2(3_1)=a_2(3_1)=2, which
    implies that
    
    Z_2(Wh(3_1)) = Z_2(Wh(3_1))  .
    
    On the other hand, V_Wh(3_1),2(t,q) ≠ V_Wh(3_1),2(t,q),
    the exact values given in Equation (<ref>) in the Appendix. This implies the
    following. 
    
    
    The V_2-polynomial is not determined by the 2-loop part of the Kontsevich integral.   
    
    
    Based on limited computations available, it was observed in <cit.> that
    Khovanov Homology alone, or HFK alone, or the colored Jones polynomial alone
    do not determine V_2. 
    
    
    
     §.§ A relation between V_1 and V_2
    
    
    In a sense, the sequence of V_n-polynomials are similar to the sequence of the
    Jones polynomial of a knot <cit.> and its (n,1)-parallels. In fact, it
    follows from the axioms of the TQFT that the Jones polynomial of a parallel of
    knot is a linear combination (with coefficients that are independent of the knot) of 
    colored Jones polynomial, colored by the irreducible representations of
    𝔰𝔩_2(); see <cit.>, and vice-versa.
    
    It was recently conjectured in <cit.> that the V_n-polynomials of a knot
    are also linear combinations of the V_1-polynomial of a knot and its (n',1)-cables
    for n' ≤ n, and a proof for n=2 was given there. We illustrate this relation
    here, and at the same time giving a consistency between the coefficients of the
    relation computed by the spectral decomposition of R-matrices and by
    representation theory in <cit.> with the computer-program that computes V_1
    and V_2. The following relation holds for the unknot,
    3_1, 4_1, 6_1, 6_2, 6_3, 7_7, 8_3, 8_4 and their mirrors (in total,
    14 knots)
    
    
    V_K,2(t^2,q^2) = c_2,0(t,q) V_K(2,1),1(t,q)
    + c_2,-1(t,q) V_K,1(t^2 q^-1,q)
    + c_2,1(t,q) V_K,1(t^2 q,q)  ,
    
    where
    
    c_2,-1(t,q) = t(t^2 q^2 -1)/q(1+q^2)(t^2-1),     
    c_2,1(t,q) = t^2-q^2/qt(1+q^2)(t^2-1),     
    c_2,0(t,q) = (q+t)(1+qt)/(1+q^2)t
    satisfy the symmetry c_2,1(t,q) = c_2,-1(t^-1,q),
    c_2,0(t,q)=c_2,0(t^-1,q).
    
    
    
    
    Some values of (<ref>) are given in the appendix.
    
    
    
     §.§ The V_2-polynomial of torus knots with two strings
    
    
    From the general setting, it follows that if β and γ are elements of
    a braid group of a fixed numbers of strands and K_n denotes the link obtained 
    by the closure of β^n γ, then K_n is a knot if n lies in an arithmetic
    progression and the sequence V_1(K_n)(t,q) is holonomic and satisfies a linear
    recursion relation with coefficients in [t^± 1,q^± 1]) coming from the
    minimal
    polynomial of the square of the R-matrix. This can be computed explicitly and
    leads to the answer. The above holds locally, if we replace tangle in a planar
    projection of a knot by β^n γ, and holds for any of the polynomial
    invariants that we discuss in this paper.
    
    We illustrate this giving a recursion relation of the values of 
    V_1 and V_2 for T(2,2b+1)-torus knots for an integer b.
    The minimal polynomial of the square of the R-matrix of V_1 is
    
    (-1 + x) (-t^2 + q^2 x) (-1 + q^2 t^2 x) =
    -t^2 + (q^2 + t^2 + q^2 t^4) x -(q^2 + q^4 t^2 + q^2 t^4) x^2 + q^4 t^2 x^3  .
    
    It follows that f_b(t,q) = V_T(2,2b+1),1(t,q) satisfies the recursion relation
    
    -t^2 f_b(t,q) + (q^2 + t^2 + q^2 t^4) f_b+1(t,q)
    -(q^2 + q^4 t^2 + q^2 t^4) f_b+2(t,q) + q^4 t^2 f_b+3(t,q) =0
    
    for b ∈ with initial conditions
    
    f_-1(t,q)=1,    f_0(t,q)=1,    f_1(t,q) =
    1 + (q^-1+q^-3) u + q^2 u^2
    
    where
    
    u=t+t^-1-q-q^-1 .
    
    
    
    This and the t ↔ t^-1 symmetry of V_1 implies that
    f_b(t,q) = q^-2b (t^2b+t^-2b) + (lower order terms)
    for b ≥ 0, thus _t(f_b(t,q))=4b=4 ·genus(T(2,2b+1)) for
    b>0. It follows that inequality (<ref>) for n=2 is an equality
    for b ≥ 0. Since T(2,2b+1)=T(2,-2b-1), it follows that
    f_b(t,q^-1)=f_-b-1(t,q) which then concludes that inequality (<ref>)
    for n=1 is an equality for all 2-string torus knots.
    
    Likewise, the minimal polynomial of the square of the R-matrix of V_2 is
    
    (-1 + x) (-t^2 + q^2 x) (-1 + q^3 x) (-t^2 + q^4 x) (-1 + q^2 t^2 x) (-1 + q^4 t^2 x)
    
    
    
    which translates into a 6th order linear recursion relation for 
    g_b(t,q) = V_T(2,2b+1),2(t,q) with initial conditions 
    
    
    g_-1(t,q) =     g_0(t,q^-1) = 1 
    
    g_-2(t,q) =     g_1(t,q^-1) =
    1 + (q + 2 q^3 - q^4 + q^5 - q^6) u + (q^2 + q^4 - q^5) u^2
    
    
    g_-3(t,q) =     g_2(t,q^-1) =
    1 + (2 q + 3 q^3 - q^4 + 3 q^5 - q^6 + 2 q^7 - q^8 + q^9 - 2 q^10 + 
    q^11 - q^12) u 
        + (4 q^2 + 7 q^4 - 3 q^5 + 10 q^6 - 6 q^7 + 
    6 q^8 - 7 q^9 + 3 q^10 - 3 q^11) u^2 
        + (3 q^3 + 6 q^5 - 3 q^6 + 6 q^7 - 6 q^8 + 3 q^9 - 3 q^10) u^3
    + (q^4 + q^6 - q^7 + q^8 - q^9) u^4
    
    with u as in (<ref>). As in the case of V_1, from the above recursion
    one deduces that inequality (<ref>) for n=2 is in fact an equality
    for all 2-string torus knots. 
    
    We have performed the analogous calculation for the case of the V_3 and V_4
    polynomials, and the conclusion is that inequality (<ref>) for n=1, …, 4
    is in fact an equality for all 2-string torus knots. 
    
    
    
     §.§ Positivity of the V_1 and V_2-polynomials of
    alternating knots?
    
    
    The next topic that we discuss is a curious positivity observation for the
    coefficients of the V_1 and V_2 polynomials of alternating knots.
    Recall the number of alternating knots with at most 15 crossings
    (up to mirror image) given in Table <ref> and taken from <cit.>.
    
    
    
    After computing the V_1 and V_2 polynomials in the following range of knots,
    we observed the following.
    
    
    For all alternating knots with ≤ 15 crossings, we have
    
    V_1(t,-q) ∈_≥ 0[t^± 1,q^± 1]  .
    
    The same conclusion holds for V_2 for all alternating knots with ≤ 14
    crossings.
    
    
    The above positivity fails for V_3(t,-q) and V_4(t,-q) already both for the
    3_1 and the 4_1 knots.
    
    
    Is this an accident of knots with low number of crossings or a hint of a
    relation of V_1 and V_2 with some categorification theory?
    
    
    
    § COMPUTING THE V_N-POLYNOMIALS
    
    
    A priori, the polynomial invariant of long knots based on an R-matrix on a
    d-dimensional vector space is a state sum of d^2c terms where c is the
    number of crossings of a planar projection of a knot knot, and in the case of
    the V_n polynomials, d=4n. Even though the summand is sparse, a direct
    computation of the V_2 polynomial for knots with 8 crossings is unfeasible. 
    A key observation is that every polynomial invariant of long knots of <cit.>
    is given as a state sum and has a natural local tangle version due to the fact
    that oriented crossings are allowed to be oriented up/downwards but also sideways.
    The locality property of this polynomial is very important for its efficient
    computation, an idea that highlighted time and again in the work of Bar-Natan and
    van der Veen.
    
    Given a planar diagram of a knot, the computation is assembled from the following
    parts:
    
      * 
      Convert the planar diagram into the planar diagram of a corresponding long knot,
      with up-pointing crossings, and record the rotation number for each arc;
    
      * 
      Correspond each crossing with the pre-computed R-matrices. Since the crossings
      are up-pointing, there are only two kinds of crossings (positive and negative);
    
      * 
      Tensor contract the crossings until the polynomial is obtained, with respect to a
      specific order that simplifies the computation. The contraction is done accordingly
      to the rotation numbers of arcs, which, together with the pre-computed curls,
      determine the signs of indeterminate terms.
    
    
    Part (1) is done by directly calling the function  from <cit.>.
    The part most critical to the speed of computation is to determine the order to which
    the contraction is performed. Before explaining our method of determining the order,
    we briefly review some standard terminologies of tensor contraction. 
    
    An n-tensor is a tuple indexed by a set of the form 
    
        { 1,⋯,m_1 }×⋯×{ 1,⋯ m_n },
    
    where, for k∈{ 1,⋯,n }, the integer m_k is the dimension
    of the legk of the tensor. 
    
    For an n-tensor T, the entry corresponding to an element
    { i_1,⋯,i_n } in the index set is denoted as T_i_1,⋯ ,i_n.
    Given two tensors T and T', if T has a leg k whose dimension m_k is the
    same as a leg k' of T', we can contractT and T' along the pair of
    legs k and k' to a new tensor T”, defined by
    
        T”_i_1,⋯,i_k,⋯,i_n,j_1,⋯,j_k',⋯,j_n'∑_i_k, j_k'∈{ 1,⋯,m_k }
        
        i_k = j_k' T_i_1,⋯,i_k,⋯,i_nT'_j_1,⋯,j_k',⋯,j_n',
    
    where the hats indicate that the indices under them are deleted. Similarly we can
    define contractions of contracting multiple pairs of legs and/or more than two tensors
    at once. By definition, assuming each multiplication and addition is of the same
    complexity (and ignoring the fact that we are adding and multiplying polynomials
    with integer coefficients in two variables, rather than integers), the time
    complexity of a contraction is described by
    
        Product of dimensions of all legs of tensors
            involved/Product of dimensions of all pairs of legs contracted.
    
    For example, the time complexity of contracting two 3-tensors along two pairs of
    legs into a 2-tensor, where all legs are of dimension d, is
    d^3· d^3/d· d = d^4. 
    
    In our case, the tensors come from the R-matrices, which are indexed as 4-tensors
    with 4n-dimensional legs for the V_n-polynomial. When associating the crossings
    with R-matrices in part (2), the four arcs of the crossings are also associated to
    the four legs of the corresponding 4-tensors according to a carefully arranged order. 
    Tensor contraction of crossings in part (3) means to contract the associated tensors
    along their common legs (i.e. legs associated to same arcs) up to the sign provided
    by the curls and rotation numbers. After all arcs, except the entrance and exit arcs,
    have been contracted, we obtain a 2-tensor which is a 4n× 4n diagonal matrix
    with identical diagonal entries which give the desired V_n-polynomial. This allows
    us to reduce the legs associated to the entrance and exit arcs to only one dimension
    before computing the contractions. 
    
    When all dimensions of legs are equal, taking the logarithm with the dimension as
    the base to (<ref>) gives the following description of the time complexity
    
        log_dim(time complexity) = #{legs of tensors involved}
        - #{pairs of legs contracted},
    
    where # S stands for the cardinality of a set S. 
    
    Therefore, to reduce the time complexity, we need to reduce the number of legs of
    tensors involved and increase the number of pairs of legs contracted per contraction.
    For our case, this means that we contract all contractible legs of two tensors at a
    time, and find an order of contractions so that the right hand side of (<ref>)
    is as small as possible. The latter is the most difficult, and is well-known to be
    NP-hard for general graphs (instead of only the planar diagrams of knots) in the field
    of tensor network. Due to the less complex nature of planar diagrams of knots, we do
    not need to go as far as NP-hard, while being able to reduce the time complexity to a
    satisfactory level with some techniques. One approach to this is to prioritize
    contracting bigons as described in <cit.>, but we are taking a different
    approach here, described as the following: 
    
    
      * 
      Input: a planar diagram of a long knot with rotation numbers obtained by
       and R-matrices associated;
    
      * 
      While there are arcs other than the entrance and exit arcs remaining:
      
      * 
        Find all pairs of crossings with common arcs (crossings may appear multiple
        times in different pairs);
      
      * 
        Evaluate the time complexity of contracting each pair of crossings in (a) along
        all their common arcs according to (<ref>) (the entrance and exit arcs
        do not count);
      
      * 
        Contract the pair with the minimal time complexity (if there are multiple pairs
        with the minimal complexity, choose an arbitrary one).
      
      * 
      Output: Only the entrance and exit arcs remain and the desired V_n-polynomial
      is obtained.
    
    
    In practice, the time complexity depends on which arc is split when the knot is
    converted into a long knot, so before actually performing the computations of
    R-matrices, we execute the above algorithm with each possible long knot as the
    input, contract formally without computing the R-matrices, record the time
    complexities in each step (c) and return the maximal time complexity recorded as
    the estimated complexity for the input long knot. After all these, we perform the
    actual computation on the long knot with the minimal estimated complexity. 
    
    For example, it takes a few minutes to compute the V_2-polynomial of the loose
    knots with 11 crossings from (<ref>).
    
    
    
    
     §.§ Acknowledgements
    
    
    The authors wish to thank Dror Bar-Natan, Nathan Dunfield, Rinat Kashaev,
    Ben-Michael Kohli, Ciprian Manolescu, Mingde Ren and Roland van der Veen
    for useful conversations. 
    
    
    
    
    § V_2-EQUIVALENCE CLASSES OF KNOTS WITH AT MOST 14 CROSSINGS
    
    
    In this section we list the V_2-equivalence classes (up to mirror image)
    of knots with at most 14 crossings. As perhaps expected, the equivalence classes
    involve knots with the same number of crossings. Overline means mirror image. 
    The counts are given in Table <ref>.
    
    
    
    
    
    
    
    
    Below, we indicate the flavor of each equivalence class (defined in
    Equation (<ref>)) by the corresponding number in the superscript.
    There are 218 tuples in total, 172 being tight + thin, 30 tight + thick,
    and 16 loose + thick. 
      
    First, we give the 3 pairs with 12 crossings.
    
    
    (12n364,    12n365)^1        
    (12n421,    12n422)^1        
    (12n553,    12n556)^1  
    
    
    Next, we give the 25 pairs of knots with 13 crossings.
    
    
      (13a141,   13a142)^1       
      (13a199,   13a204)^1       
      (13a906,   13a916)^1       
      (13a1114,   13a1143)^1 
    
      (13a1126,   13a1163)^1       
      (13a1813,   13a1831)^1       
      (13a1991,   13a2021)^1       
      (13a1995,   13a2006)^1 
    
      (13a2720,   13a2727)^1       
      (13a2802,   13a2808)^1       
      (13n370,   13n373)^1       
      (13n372,   13n375)^3 
    
      (13n404,   13n416)^1       
      (13n406,   13n418)^1       
      (13n534,   13n549)^1       
      (13n536,   13n551)^3 
    
      (13n875,   13n950)^1       
      (13n1129,   13n1130)^1       
      (13n1653,   13n1683)^3       
      (13n1655,   13n1685)^2 
    
      (13n1894,   13n2099)^1       
      (13n2185,   13n2229)^1       
      (13n2205,   13n2250)^3       
      (13n2933,   13n2956)^1 
    
       (13n2937,   13n2955)^1                   
    
    Next, we give the 189 pairs and one triple of knots with 14 crossings.
    
    
      (14a34,  14a35)^1       
      (14a96,  14a103)^1       
      (14a454,  14a458)^1       
      (14a518,  14a592)^1 
    
      (14a533,  14a550)^1       
      (14a608,  14a617)^1       
      (14a675,  14a734)^1       
      (14a718,  14a736)^1 
    
      (14a989,  14a1017)^1       
      (14a1047,  14a1170)^1       
      (14a1268,  14a1362)^1       
      (14a1445,  14a1449)^1 
    
      (14a1522,  14a1532)^1       
      (14a1767,  14a1860)^1       
      (14a2205,  14a2215)^1       
      (14a2253,  14a2256)^1 
    
      (14a2609,  14a2618)^1       
      (14a3400,  14a3433)^1       
      (14a3403,  14a3432)^1       
      (14a3407,  14a3436)^1 
    
      (14a3409,  14a3438)^1       
      (14a3419,  14a3439)^1       
      (14a4041,  14a4998)^1       
      (14a4147,  14a4939)^1 
    
      (14a4901,  14a5698)^1       
      (14a6467,  14a6614)^1       
      (14a6614,  14a6467)^1       
      (14a7193,  14a7216)^1 
    
      (14a7196,  14a7269)^1       
      (14a7200,  14a7249)^1       
      (14a7207,  14a7272)^1       
      (14a7210,  14a7275)^1 
    
      (14a7215,  14a7698)^1       
      (14a7219,  14a7260)^1       
      (14a7258,  14a7263)^1       
      (14a7264,  14a7274)^1 
    
      (14a7446,  14a7477)^1       
      (14a7527,  14a7598)^1       
      (14a8017,  14a8107)^1       
      (14a8096,  14a8115)^1 
    
      (14a9707,  14a9711)^1       
      (14a10116,  14a10142)^1       
      (14a10142,  14a10116)^1       
      (14a10405,  14a10410)^1 
    
      (14a10407,  14a10434)^1       
      (14a10411,  14a10453)^1       
      (14a10414,  14a10417)^1       
      (14a10415,  14a10435)^1 
    
      (14a10439,  14a10456)^1       
      (14a10853,  14a11544)^1       
      (14a11793,  14a12325)^1       
      (14a12335,  14a12344)^1 
    
      (14a12816,  14a12833)^1       
      (14a12868,  14a12876)^1       
      (14a13431,  14a13433)^1       
      (14a13473,  14a13475)^1 
    
      (14n179,  14n182)^1       
      (14n181,  14n184)^3       
      (14n213,  14n226)^1       
      (14n215,  14n228)^1 
    
      (14n364,  14n386)^1       
      (14n366,  14n388)^3       
      (14n733,  14n810)^1       
      (14n1366,  14n1393)^1 
    
      (14n1370,  14n1395)^2       
      (14n1374,  14n1397)^1       
      (14n1378,  14n1399)^1       
      (14n1380,  14n1401)^3 
    
      (14n1692,  14n1704)^1       
      (14n1697,  14n1945)^1       
      (14n1699,  14n1947)^2       
      (14n1701,  14n1949)^1 
    
      (14n1752,  14n1753)^1       
      (14n1762,  14n1839)^2       
      (14n1764,  14n1841)^2       
      (14n1766,  14n1843)^2 
    
      (14n1768,  14n1775)^1       
      (14n1770,  14n1835)^1       
      (14n1772,  14n1837)^1       
      (14n2007,  14n2032)^2 
    
      (14n2148,  14n2372)^1       
      (14n2150,  14n2374)^3       
      (14n2295,  14n2376)^1       
      (14n2297,  14n2378)^1 
    
      (14n2299,  14n2380)^2       
      (14n2423,  14n5868)^2       
      (14n3418,  14n3823)^1       
      (14n3422,  14n3825)^3 
    
      (14n3444,  14n3836)^1       
      (14n3448,  14n3829)^1       
      (14n3450,  14n3831)^2       
      (14n4577,  14n4583)^2 
    
      (14n4579,  14n4585)^2       
      (14n4657,  14n4665)^1       
      (14n4925,  14n5085)^1       
      (14n4930,  14n4931)^1 
    
      (14n4933,  14n5087)^2       
      (14n4938,  14n4939)^1       
      (14n5822,  14n5852)^3       
      (14n5854,  14n5862)^1 
    
      (14n7506,  14n7559)^1       
      (14n7566,  14n7673)^1       
      (14n7575,  14n7675)^2       
      (14n7577,  14n7677)^2 
    
      (14n7586,  14n7678)^1       
      (14n7593,  14n7680)^1       
      (14n7597,  14n7682)^2       
      (14n7599,  14n7684)^2 
    
      (14n7603,  14n7686)^1       
      (14n7617,  14n7628)^2       
      (14n7636,  14n7688)^1       
      (14n7638,  14n7690)^1 
    
      (14n8225,  14n10806)^2       
      (14n8291,  14n8293)^1       
      (14n8648,  14n8649)^1       
      (14n8650,  14n8651)^1 
    
      (14n8696,  14n8697)^1       
      (14n9075,  14n9076)^1       
      (14n9139,  14n9140)^1       
      (14n9142,  14n9143)^1 
    
      (14n9395,  14n9396)^1       
      (14n9398,  14n9399)^1       
      (14n9455,  14n9456)^1       
      (14n9458,  14n9459)^1 
    
      (14n9686,  14n9687)^1       
      (14n10002,  14n11740)^3       
      (14n10503,  14n11641)^3       
      (14n11679,  14n11981)^3 
    
      (14n14122,  14n14288)^1       
      (14n14130,  14n14216)^1       
      (14n14134,  14n14215)^2       
      (14n14136,  14n14219)^3 
    
      (14n14148,  14n14150)^1       
      (14n14149,  14n14151)^1       
      (14n14154,  14n14328)^1       
      (14n14156,  14n14157)^1 
    
      (14n14158,  14n14159)^1       
      (14n14162,  14n14169)^1       
      (14n14177,  14n14330)^3       
      (14n14196,  14n14333)^1 
    
      (14n14204,  14n14341)^1       
      (14n14208,  14n15068)^2       
      (14n14210,  14n14225)^1       
      (14n14214,  14n14131)^2 
     
      (14n14223,  14n14211)^2       
      (14n14227,  14n14335)^1       
      (14n14313,  14n14319)^3       
      (14n14322,  14n14332)^2 
    
      (14n14504,  14n14506)^1       
      (14n14508,  14n14502)^1       
      (14n14511,  14n14513)^1       
      (14n14516,  14n14509)^1 
    
      (14n14590,  14n14663)^1       
      (14n14655,  14n14685)^1       
      (14n14687,  14n15694)^1       
      (14n14780,  14n14893)^2 
    
      (14n14787,  14n14895)^1       
      (14n14793,  14n14897)^1       
      (14n14808,  14n14804)^1        
      (14n14924,  14n14923)^1
    
      (14n14926,  14n14925)^1       
      (14n14931,  14n14930)^1       
      (14n15024,  14n15022)^1       
      (14n15059,  14n15058)^1 
    
      (14n15063,  14n15062)^1       
      (14n15066,  14n15065)^1       
      (14n15084,  14n15083)^1       
      (14n15103,  14n15102)^1 
    
      (14n15106,  14n15105)^1       
      (14n15173,  14n15172)^1       
      (14n15180,  14n15179)^1       
      (14n15202,  14n15201)^1 
    
      (14n15205,  14n15204)^1       
      (14n15208,  14n15207)^1       
      (14n15228,  14n15227)^1       
      (14n15232,  14n15231)^1 
    
      (14n15258,  14n15257)^1       
      (14n15727,  14n15756)^1       
      (14n15729,  14n15758)^2       
      (14n17934,  14n17940)^1 
    
      (14n17938,  14n17936)^2       
      (14n17942,  14n17946)^1       
      (14n17949,  14n17960)^2       
      (14n17986,  14n18013)^1 
    
      (14n17997,  14n18017)^1       
      (14n18005,  14n18012)^2       
      (14n18146,  14n18144)^1       
      (14n18208,  14n18207)^1 
    
      (14n19744,  14n19758)^2 
                        
      (14n14212,  14n14213,  14n14222)^1
    
    
    
    
    
    
    
    The knots in Tables (<ref>),  (<ref>),
     (<ref>)–(<ref>) are Conway mutant, have equal HKF and
    Equation (<ref>) is an equality for n=2. The knots in
    Tables (<ref>),  (<ref>) have equal V_1, V_2, V_3 and V_4
    polynomials and the ones in Tables  (<ref>)–(<ref>) have equal
    V_1, V_2 and V_3 polynomials with the two exceptions as in the footnote
    of Proposition <ref>.
    
    All but at most 12 of the tight + thin knots listed above are quasi-alternating knots.
    We examined them with a computer search, extending further the table
    of quasi-alternating knots with ≤ 12 crossings of Jablan <cit.>.
    All but the following 12 tight + thin knots were confirmed to be
    quasi-alternating:
    
        14n2378,  14n3448,  14n4925,    14n5085,  14n5862,  14n5854,
        
          14n7559,  14n7506,  14n14151,    14n14149,  14n14169,  14n14162  .
    
      
    
    
    
    § VALUES FOR WHITEHEAD DOUBLES AND (2,1)-PARALLELS
    
    
    The values of V_Wh(K),2(t,q) for the first three nontrivial knots
    is given as follows, where u=t+t^-1-q-q^-1 is as in (<ref>). 
    
    
    
    V_Wh(3_1),2(t,q) =    
      1 + (-2 - 2 q^-2 + 2 q^-1 + 4 q - 4 q^2 + 4 q^3 - 2 q^4 - 2 q^7 + 2 q^8 - 
      2 q^10 + 2 q^11 + 2 q^15 - 2 q^16 + 2 q^17
        - 2 q^18
     - 2 q^20 + 
        4 q^22 - 2 q^23) u + (2 + 2 q^-2 - 2 q^-1 - 4 q + 2 q^2 - 4 q^3 + 
        2 q^4 + 4 q^5 - 2 q^6 + 4 q^7 
        - 4 q^8 + 4 q^9   - 6 q^10
    + 2 q^13 - 
        2 q^14 + 2 q^15 + 2 q^18 - 2 q^19 + 2 q^20 - 4 q^21 + 2 q^22) u^2,
    
        
    V_Wh(3_1),2(t,q) =    
    1 + (-2 q^-26 + 4 q^-25 - 2 q^-23 - 2 q^-21 + 2 q^-20 - 2 q^-19
    + 2 q^-18 + 2 q^-14 - 2 q^-13 + 2 q^-11
        - 2 q^-10 + 2 q^-9
    - 4 q^-8 + 4 q^-7 - 6 q^-6 + 4 q^-5 - 4 q^-4 + 4 q^-3) u
    + (- 2 q^-25 + 4 q^-24 - 2 q^-23
        + 2 q^-22 - 2 q^-21 - 2 q^-18
    + 2 q^-17 - 2 q^-16 + 6 q^-13 - 4 q^-12 + 4 q^-11 - 4 q^-10
    + 2 q^-9 - 2 q^-8
        - 2 q^-7 + 4 q^-6 - 4 q^-5 
    + 2 q^-4 - 2 q^-3 + 2 q^-2) u^2
    
    and for fun,
    
    
    V_Wh(4_1),2(t,q) =    
    1 + (-14 - 2 q^-18 + 4 q^-17 + 2 q^-16 - 6 q^-15 - 4 q^-14
    + 6 q^-13
    + 6 q^-12 - 4 q^-11 - 8 q^-10 + 4 q^-9
        + 8 q^-8 - 4 q^-7
    - 10 q^-6 + 16 q^-5 - 16 q^-4 + 20 q^-3 - 22 q^-2 + 18 q^-1
    + 8 q + 2 q^2 - 4 q^3 - 6 q^4 + 8 q^5 
        + 4 q^6 - 8 q^7 - 4 q^8 + 6 q^9
    + 6 q^10 - 4 q^11 - 6 q^12 + 2 q^13 + 4 q^14 - 2 q^15) u
    + (24 - 2 q^-17 + 4 q^-16
        - 2 q^-14 - 4 q^-13 + 4 q^-12
    + 2 q^-11 + 4 q^-10 - 16 q^-9 + 10 q^-8 + 8 q^-7 - 10 q^-6
    - 4 q^-5 + 20 q^-4
        - 28 q^-3 + 28 q^-2 - 28 q^-1 - 16 q + 
        2 q^2 + 12 q^3 - 8 q^4 - 10 q^5 + 16 q^6 - 4 q^7 - 2 q^8 - 
        4 q^9 + 4 q^10 + 2 q^11
        - 4 q^13 + 2 q^14) u^2
    
    with u as in (<ref>).
    
    
    
    
    § VALUES FOR  (2,1)-PARALLELS OF KNOTS
    
    
    We now give values of the V_K,1, V_K(2,1),1 and V_K,2 for some sample
    knots to explicitly confirm Equation (<ref>).
    
    
    
    V_3_1,1(t,q) =    
    1 + (q + q^3) u + q^2 u^2,
    
      
    V_3_1,2(t,q) =    
    1 + (q + 2 q^3 - q^4 + q^5 - q^6) u + (q^2 + q^4 - q^5) u^2,
    
      
    V_3_1(2,1),1(t,q) =    
     1 + (q^-3 + 2 q + 3 q^3 + 2 q^5 + q^7 - q^13) u + (3 + 3 q^-2 + 
     6 q^2 + 4 q^4 + 4 q^6 + 2 q^8 - 2 q^10 - 2 q^12) u^2 
       
     + (3 q^-1 + 3 q + q^3 + q^5 + q^7 - q^11) u^3 + u^4
    
    V_4_1,1(t,q) =    
    1 + (- q^-1 - q) u + u^2,
    
      
    V_4_1,2(t,q) =    
    1 + (2 - q^-3 + q^-2 - 2 q^-1 - 2 q + q^2 - q^3) u
    + (1 + q^-2 - q^-1 - q + q^2) u^2, 
    
      
    V_4_1(2,1),1(t,q) =    
    1 + (- q^-7 + q^-5 - 3 q^-3 - 3 q^-1 - q^3 - q^7) u
    + (2 + 2 q^-6 + 5 q^-4 + q^-2 + 2 q^4 + 2 q^6) u^2 
       
    + (q^-5 + 3 q^-3 + 3 q^-1 + q^5) u^3 + q^-2 u^4  
    
    V_6_1,1(t,q) =    
    1 + (- q^-1 - 2 q - q^3) u + (3 + q^2) u^2,
    
      
    V_6_1,2(t,q) =    
     1 + (1 - q^-3 - 2 q^-1 - 2 q + 2 q^2 - 2 q^3 + 2 q^4 - 2 q^5 + q^6 - 
     q^7) u 
        
     + (5 + 3 q^-2 - 2 q^-1 - 5 q + 3 q^2 - 3 q^3 + 3 q^4 - q^5 + q^6) u^2, 
    
      
    V_6_1(2,1),1(t,q) =    
    1 + (- q^-7 + 2 q^-5 - 5 q^-3 - 8 q^-1 - 2 q - q^11 - q^15) u
    + (15 + 6 q^-6 + 17 q^-4 + 16 q^-2 - 2 q^2 - 4 q^4 + 4 q^8 
        + 4 q^10 + 
    2 q^12 + 2 q^14) u^2
    + (3 q^-5 + 10 q^-3 + 15 q^-1 + 3 q - 2 q^3 + 2 q^9 + q^13) u^3
    + (1 + 3 q^-2) u^4  
    
    V_6_2,1(t,q) =    
    1 + (- q^-1 - q^5) u + (1 - q^2 - q^4) u^2 + (q + q^3) u^3 + q^2 u^4,
    
      
    V_6_2,2(t,q) =    
    1 + (1 - q^-3 + q^-2 - 2 q^-1 - q + q^2 - q^5 + q^6 - 2 q^7 + 2 q^8 - q^9) u
    + (-1 + q^-2 - q^-1 + 2 q - 4 q^2 + 5 q^3 
       
    - 6 q^4 + 5 q^5 - 3 q^6 + 2 q^7 - q^8) u^2
    + (-1 + q^-1 + 3 q - 3 q^2 + 4 q^3 - 4 q^4 + 3 q^5 - 2 q^6 + q^7) u^3 
       
    + (1 - q + 2 q^2 - 2 q^3 + 2 q^4 - 2 q^5 + q^6) u^4, 
    
      
    V_6_2(2,1),1(t,q) =    
     1 + (q^-9 - 3 q^-7 + q^-3 - 6 q^-1 + 4 q^3 - 2 q^5 - 2 q^7 - q^13 + 
     q^17 - q^19) u
     + (7 q^-8 - q^-6 - 5 q^-4 - q^-2 - 8 q^2 
        
     - 12 q^4 - q^6 + q^8 + 2 q^10 + 2 q^12 - 2 q^18) u^2
     + (21 q^-7 + 20 q^-5 + 6 q^-3 + 10 q^-1 + 18 q + 14 q^3 + 18 q^5
     + 9 q^7 
        - 5 q^9 + 2 q^11 - q^13 + 5 q^15 + 3 q^17) u^3
     + (64 + 35 q^-6 + 62 q^-4 + 64 q^-2 + 63 q^2 + 41 q^4 + 22 q^6
     - 12 q^8 
        - 16 q^10 + 16 q^14 + 12 q^16) u^4
     + (35 q^-5 + 73 q^-3 + 74 q^-1 + 51 q + 29 q^3 + 18 q^5 - 13 q^9
     - 11 q^11 + 11 q^13
        + 13 q^15) u^5
     + (25 + 21 q^-4 + 39 q^-2 + 6 q^2 + 7 q^4 - 6 q^10 + 6 q^14) u^6
     + (7 q^-3 + 8 q^-1 + q^3 - q^11 + q^13) u^7 + q^-2 u^8  
    
    and finally, 
    
    V_8_4,1(t,q) =    
    1 + (- q^-3 - 2 q^-1 - q - q^3 - q^5) u
    + (3 + q^-2 + q^4) u^2
    + (q^-1 + 6 q + 5 q^3) u^3 + (1 + 3 q^2) u^4,
    
      
    V_8_4,2(t,q) =      
    1 + (2 - q^-7 + q^-6 - 2 q^-5 + q^-4 - 2 q^-3 + q^-2
    - 3 q^-1 - 2 q + 3 q^2 - 2 q^3 + q^4 - 2 q^5 - q^7 + q^8 - q^9) u 
       
    + (8 + q^-6 - q^-5 + 3 q^-4 - 2 q^-3 + 3 q^-2 - 5 q^-1
    - 7 q + 8 q^2 - 10 q^3 + 8 q^4 - 5 q^5 + 6 q^6 - 3 q^7 + q^8) u^2 
        
    + (-20 + q^-5 - q^-4 + 7 q^-3 - 7 q^-2 + 16 q^-1 + 24 q
    - 27 q^2 + 27 q^3 - 22 q^4 + 18 q^5 - 9 q^6 + 5 q^7) u^3 
        
    + (6 + q^-4 - q^-3 + 4 q^-2 - 4 q^-1 - 8 q + 8 q^2 - 7 q^3 + 7 q^4
    - 5 q^5 + 3 q^6) u^4, 
    
    
    V_8_4(2,1),1(t,q) =    
    1 + (-q^-15 - 5 q^-7 - 3 q^-3 - 11 q^-1 + 2 q + 6 q^3 - 6 q^5 - 
    4 q^7 + 2 q^9 - 2 q^11 - q^13 - q^19) u
    + (14 + 2 q^-14
        + 2 q^-12 + 11 q^-10 + 24 q^-8 - 2 q^-6
    - 9 q^-4 + 14 q^-2 - 6 q^2 - 11 q^4 + 17 q^8 + 8 q^10 + 8 q^12
    + 2 q^14 - 2 q^16
        + 2 q^18) u^2
    + (5 q^-13 + 8 q^-11 + 51 q^-9 + 127 q^-7 + 124 q^-5 + 89 q^-3
    + 63 q^-1 + 56 q + 46 q^3 + 49 q^5 + 52 q^7 
        + 27 q^9 + 41 q^11
    + 27 q^13 + 22 q^15 + 21 q^17) u^3
    + (141 + 12 q^-12 + 28 q^-10 + 107 q^-8 + 259 q^-6 + 330 q^-4
        
    + 255 q^-2 + 84 q^2 + 47 q^4 + 90 q^6 + 44 q^8 + 44 q^10 + 60 q^12
    + 60 q^14 + 44 q^16) u^4
    + (13 q^-11 + 24 q^-9 + 89 q^-7
        + 237 q^-5 + 309 q^-3 + 215 q^-1
    + 70 q + 5 q^3 + 39 q^5 + 41 q^7 + 20 q^9 + 21 q^11 + 44 q^13 + 41 q^15) u^5
    + (59 
        + 6 q^-10 + 6 q^-8 + 39 q^-6 + 114 q^-4 + 132 q^-2
    - 11 q^2 + 9 q^4 + 6 q^6 + 12 q^8 + 6 q^12 + 18 q^14) u^6 
       
    + (q^-9 + 10 q^-5 + 28 q^-3 + 23 q^-1 - 2 q + q^5 + 2 q^9 - 2 q^11
    + 3 q^13) u^7 + (q^-4 + 3 q^-2) u^8  
    
    
    Keep in mind that the genus of the (2,1)-parallel of K is twice the genus of
    K, and that the knots 3_1, 4_1, 6_1, 6_2 and 8_4 have genus
    1, 1, 1, 2, 2, hence we expect (and we find) that the V_1-polynomial of their
    (2,1)-parallel to have u-degree 2, 2, 2, 4, 4 conforming the equality
    in (<ref>) for the (2,1)-parallels of 3_1, 4_1, 6_1, 6_2 and 8_4.
    
    
    GS:VnDatahamsalpha
