$\mathfrak{gl}(1 \vert 1)$-Alexander polynomial for $3$-manifolds, Reidemeister torsion and lens spaces

TL;DR

This paper explores a reformulation of a 3-manifold invariant related to the $rak{gl}(1|1)$-Alexander polynomial, connecting it with Reidemeister torsion, and applies it to classify lens spaces.

Contribution

It introduces a new formulation of the invariant $ riangle(M, oldsymbol{ ho})$ and demonstrates its application in classifying lens spaces using this invariant.

Findings

Calculated the invariant for lens spaces

Established a relation between the invariant and Reidemeister torsion

Provided a classification method for lens spaces

Abstract

In this note, we reformulate the invariant $\Delta (M, \omega)$ that we defined before, and show its relation with Reidemeister torsion. We calculate $\Delta (M, \omega)$ when the $3$-manifolds are lens spaces, and discuss the classification of the lens spaces using $\Delta (M, \omega)$.