Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion - Daniel López Neumann (arXiv.org, 2021)
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.