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

TL;DR

This paper introduces a new quantum link invariant generalizing the Alexander polynomial to semisimple Lie algebras, specifically establishing a connection with the $ ext{sl}_3$ case and demonstrating its ability to distinguish complex knots.

Contribution

It extends the Alexander polynomial to a multivariable invariant for any semisimple Lie algebra, with explicit focus on $ ext{sl}_3$, and explores its properties and applications in knot distinction.

Findings

$ ext{sl}_3$ invariant relates directly to the Alexander polynomial for knots.

Parameter evaluations of the invariant recover the Alexander polynomial.

The invariant distinguishes knots like the Kinoshita-Terasaka and Conway knots.

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 $\Delta_{\mathfrak{g}}$ 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 $\Delta_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $\Delta_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $\Delta_{\mathfrak{sl}_3}$ 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.