Skein theory for the Links-Gould polynomial

TL;DR

This paper develops a cubic braid-type skein theory for the Links-Gould polynomial, enabling its computation for any oriented link and establishing connections with other link invariants and properties.

Contribution

It introduces a new skein theory for the Links-Gould polynomial, showing it can evaluate all oriented links and relates it to the V_1-polynomial and other invariants.

Findings

Links-Gould polynomial can be computed via skein theory

Establishes equality between Links-Gould and V_1-polynomial

Derives properties like Alexander polynomial specialization and Seifert genus bounds

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.