Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

DOI10.2140/AGT.2022.22.2419arXiv1911.02925MaRDI QIDQ6328730

Daniel López Neumann

Publication date: 7 November 2019

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

Mathematics Subject Classification ID

Hopf algebras and their applications (16T05) Finite-type and quantum invariants, topological quantum field theories (TQFT) (57K16) Invariants of 3-manifolds (including skein modules, character varieties) (57K31)

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