We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra with its automorphism group . These are topological invariants of balanced sutured –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 is –graded, they can be extended in a canonical way to polynomial invariants. When is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured –manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.
"Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion." Algebr. Geom. Topol. 22 (5) 2419 - 2466, 2022. https://doi.org/10.2140/agt.2022.22.2419