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Skein theory for the Links-Gould polynomial

Stavros Garoufalidis, Matthew Harper, Rinat Kashaev, Ben-Michael Kohli, Jiebo Song, Guillaume Tahar

Abstract

Building further on work of Marin and Wagner, we give a cubic braid-type skein theory of the Links--Gould polynomial invariant of oriented links and prove that it can be used to evaluate any oriented link, adding this polynomial to the list of polynomial invariants that can be computed by skein theory. As a consequence, we prove that this skein theory is also shared by the $V_1$-polynomial defined by two of the authors, deducing the equality of the two link polynomials. This implies specialization properties of the $V_1$-polynomial to the Alexander polynomial and to the $\mathrm{ADO}_3$-invariant, the fact that it is a Vassiliev power series invariant, as well as a Seifert genus bound for knots.

Skein theory for the Links-Gould polynomial

Abstract

Building further on work of Marin and Wagner, we give a cubic braid-type skein theory of the Links--Gould polynomial invariant of oriented links and prove that it can be used to evaluate any oriented link, adding this polynomial to the list of polynomial invariants that can be computed by skein theory. As a consequence, we prove that this skein theory is also shared by the -polynomial defined by two of the authors, deducing the equality of the two link polynomials. This implies specialization properties of the -polynomial to the Alexander polynomial and to the -invariant, the fact that it is a Vassiliev power series invariant, as well as a Seifert genus bound for knots.

Paper Structure

This paper contains 16 sections, 21 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Link polynomials via skein theory
  3. Multivariable knot polynomials
  4. LG=V1 via skein theory
  5. Specialization, Vassiliev invariants and genus bounds for V1
  6. Organization of the paper
  7. Skein theory for the LG polynomial
  8. The n = 3 case
  9. The n = 4 case
  10. The n >= 5 case
  11. How R3 was found
  12. Basics of the LG, V1 and ADO link invariants
  13. The R-matrices for the Links--Gould, ADO and V1 polynomials
  14. The coefficients of the R3 skein relation
  15. Proof of Lemma \ref{['threestrand']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

For all links $L$ we have:

Figures (1)

  • Figure 1: The standard generators $s_i$ of the braid group $B_n$ and their inverses $\overline{s}_i$ for $i=1,\dots,n-1$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 25 more