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.
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