# Kuperberg invariants for balanced sutured 3-manifolds

## TL;DR

This paper introduces quantum invariants for balanced sutured 3-manifolds with Spin^c structures derived from involutive Hopf superalgebras, connecting them to Fox calculus and Reidemeister torsion.

## Contribution

It extends Kuperberg's construction to include non-unimodular Hopf superalgebras and incorporates Spin^c structures to define new quantum invariants.

## Key findings

- Invariant computed via Fox calculus for specific Hopf superalgebras
- Invariant relates to normalized Reidemeister torsion
- Method generalizes Kuperberg's construction with Spin^c structures

## Abstract

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 $H^{*}$.