Patterns of the $V_2$-polynomial of knots

TL;DR

This paper explores the patterns of the $V_2$-polynomial of knots, revealing genus bounds and mutation invariance across a large knot dataset, and introduces a new sequence of multivariable knot invariants derived from Nichols algebras.

Contribution

It introduces a sequence of multivariable knot polynomials from Nichols algebras and uncovers their properties, including genus bounds and mutation invariance, across extensive knot data.

Findings

Genus bound for $V_2$ is an equality for over 350 million knots.

$V_n$-polynomials show invariance under certain Conway mutations.

Patterns in $V_n$-polynomials suggest new insights into knot invariants.

Abstract

Recently, Kashaev and the first author constructed an $R$-matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin--Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, 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 $n=1,2,3,4$. This leads to the discovery of emerging patterns, including the genus bound for $V_2$ being an equality for all 352.2 million knots with at most $19$ crossings, as well as unexpected Conway mutations that seem undetected by the $V_n$-polynomials as well as by Heegaard Floer Homology and Khovanov Homology.