Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

TL;DR

This paper introduces a new method to compute Kuperberg invariants for sutured 3-manifolds using Fox calculus, extending to polynomial invariants and relating to Reidemeister torsion, with applications to exterior algebras.

Contribution

It develops a Fox calculus approach for Kuperberg invariants, extends these to polynomial invariants for graded Hopf algebras, and connects them to Reidemeister torsion in sutured manifolds.

Findings

Invariants computed via Fox calculus for certain Hopf algebras.

Polynomial extensions of invariants for graded Hopf algebras.

Refinement of Reidemeister torsion via these invariants.

Abstract

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.