$\mathfrak{gl}(1 \vert 1)$-Alexander polynomial for $3$-manifolds

TL;DR

This paper extends the construction of 3-manifold invariants using non-semisimple categories to include the $rak{gl}(1|1)$-Alexander polynomial, demonstrating its ability to distinguish certain lens spaces.

Contribution

It introduces a new 3-manifold invariant derived from Viro's $rak{gl}(1|1)$-Alexander polynomial within the framework of relative $G$-modular categories, expanding the class of computable invariants.

Findings

Invariant distinguishes homotopy equivalent lens spaces

Extension of Reshetikhin-Turaev invariants to non-semisimple categories

Demonstrates applicability to specific 3-manifolds

Abstract

As an extension of Reshetikhin and Turaev's invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semisimple and non-semisimple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro's $\mathfrak{gl}(1\vert 1)$-Alexander polynomial. We take lens spaces $L(7, 1)$ and $L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.