A Non-Abelian Generalization of the Alexander Polynomial from Quantum ${\mathfrak{sl}}_3$

Abstract

One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum ${\mathfrak{sl}}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $\hat{I}{”}_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}={\mathfrak{sl}}_3$, we establish a direct relation between $\hat{I}{”}_{{\mathfrak{sl}}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $\hat{I}{”}_{{\mathfrak{sl}}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $\hat{I}{”}_{{\mathfrak{sl}}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.