Other quantum relatives of the Alexander polynomial through the Links-Gould invariants

TL;DR

This paper explores the connections between quantum invariants of links, specifically the Alexander polynomial and Links-Gould invariants, through the lens of certain Hopf algebras, and proves a related conjecture.

Contribution

It demonstrates that specific Hopf algebras share properties with the -1 specialization of U_q gl(n|1), leading to a proof of a conjecture on Links-Gould invariants.

Findings

Shared properties of Hopf algebras with U_q gl(n|1) at -1

Proof of a conjecture relating Alexander polynomial and Links-Gould invariants

Connections between quantum algebra structures and link invariants

Abstract

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show these Hopf algebras share properties with the $-1$ specialization of $U_q gl(n|1)$ leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.