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Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

Daniel López Neumann

Algebraic & Geometric Topology 22 (2022) 2419–2466

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra H with its automorphism group  Aut(H). These are topological invariants of balanced sutured 3–manifolds endowed with a homomorphism of the fundamental group into  Aut(H), and possibly with a  Spinc structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if H is –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.

quantum invariants, Reidemeister torsion, sutured manifolds
Mathematical Subject Classification
Primary: 57K16, 57K31
Secondary: 16T05
Received: 4 November 2020
Revised: 14 April 2021
Accepted: 3 June 2021
Published: 25 October 2022
Daniel López Neumann
Institut de Mathématiques de Jussieu - Paris Rive Gauche
Université de Paris