$\mathfrak{g}\mathfrak{l}(1\mathrm{\mid }1)$-Alexander polynomial for $3$-manifolds, Reidemeister torsion and lens spaces

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