The colored Jones polynomial does not determine the Links--Gould polynomial

Abstract

We prove that the colored Jones polynomial of a knot does not determine the Links--Gould polynomial (associated with the 4-dimensional representation of the Lie superalgebra $\mathfrak{sl}(2|1)$) with two different approaches: the first is to find a concrete genus 2 mutation pair with different Links--Gould polynomials, and the second follows from the fact that the colored Jones polynomial is invariant under Conway mutation whereas the colored Links--Gould polynomial is not. The first approach also applies to a related family of polynomial invariants of Kashaev--Mangazeev, which are conjectured to be the $\mathfrak{sl}(n|1)$-knot invariants colored by the $2^n$-dimensional irreducible representation.