On the colored Links--Gould polynomial

Abstract

We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the cabling formula, we obtain genus bounds and a specialization to the Alexander polynomial for the colored Links--Gould polynomial that is independent of $n$, which implies corresponding properties of the $V_n$-polynomial for $n=2$ conjectured in previous work of two of the authors, and extends the work done for $n=1$. Combined with work of one of the authors arXiv:2409.03557, our genus bound for $\mathrm{LG}^{(2)}=V_2$ is sharp for all knots with up to $16$ crossings.