Patterns of the $V_2$-polynomial of 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.