Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles

TL;DR

This paper introduces a new class of quantum invariants for knots derived from twisted Drinfeld doubles, generalizing previous invariants and connecting them to Reidemeister torsion, with applications to $SL(n,\mathbb{C})$-twisted torsion.

Contribution

It extends twisted Kuperberg invariants to non-involutory Hopf algebras using twisted Drinfeld doubles, linking quantum invariants with Reidemeister torsion.

Findings

Reshetikhin-Turaev invariants from twisted Drinfeld doubles are defined for knots.

These invariants generalize involutory Kuperberg invariants to non-involutory cases.

The $SL(n,\mathbb{C})$-twisted Reidemeister torsion is expressed as a Reshetikhin-Turaev invariant.

Abstract

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product $\text{Rep}(H)\rtimes\text{Aut}(H)$. These are quantum invariants of knots endowed with a homomorphism of the knot group to $\text{Aut}(H)$. We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the $SL(n,\mathbb{C})$-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant.