$\mathfrak{g}\mathfrak{l}(1\mathrm{\mid }1)$-Alexander polynomial for $3$-manifolds

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 $\mathfrak{gl}(1\vert 1)$-Alexanderpolynomial\mathrm{.}Wetakelensspaces$L(7, 1)$and$L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.