Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

López Neumann, Daniel

doi:10.2140/agt.2022.22.2419

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Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

Daniel López Neumann

Algebraic & Geometric Topology 22 (2022) 2419–2466

DOI: 10.2140/agt.2022.22.2419

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 $Aut (H)$ . These are topological invariants of balanced sutured $3$ –manifolds endowed with a homomorphism of the fundamental group into $Aut (H)$ , and possibly with a ${Spin}^{c}$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $ℕ$ –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

References

Publication

Received: 4 November 2020

Revised: 14 April 2021

Accepted: 3 June 2021

Published: 25 October 2022

Authors

Daniel López Neumann
Institut de Mathématiques de Jussieu - Paris Rive Gauche Université de Paris Paris France

Volume 22, issue 5 (2022)