The Colored Jones Function does not determine any colored Links--Gould polynomial

Abstract

We prove that there is no uniform description of $\mathrm{LG}^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n=1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita--Terasaka knots by $\mathrm{LG}^{(2)}$. We develop an odd cabling formula and use a Vandermonde argument to show that for each $n\geq 2$ some satellite of each of these knots is distinguished by $\mathrm{LG}^{(n)}$.