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

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 $\mathfrak{s}\mathfrak{l}(2\mathrm{\mid }1)$) 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 $\mathfrak{s}\mathfrak{l}(n\mathrm{\mid }1)$-knot invariants colored by the $2^n$-dimensional irreducible representation.