  We construct quantum invariants of balanced sutured 3-manifolds with a
$Spin^{c}$ structure out of an involutive (possibly non-unimodular) Hopf
superalgebra $H$. If $H$ is the Borel subalgebra of
$U_{q}(\mathfrak{gl}(1|1))$, we show that our invariant is computed via Fox
calculus and it is a normalization of Reidemeister torsion. The invariant is
defined via a modification of a construction of G. Kuperberg, where we use the
$Spin^{c}$ structure to take care of the non-unimodularity of $H$ or $H^{*}$.
