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Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

### Daniel López Neumann

Algebraic & Geometric Topology 22 (2022) 2419–2466
##### 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 . These are topological invariants of balanced sutured $3$–manifolds endowed with a homomorphism of the fundamental group into , and possibly with a structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $ℕ$–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.

##### Keywords
quantum invariants, Reidemeister torsion, sutured manifolds
##### Mathematical Subject Classification
Primary: 57K16, 57K31
Secondary: 16T05