A topological interpretation of Viro's $\mathfrak{gl}(1|1)$-Alexander polynomial of a graph

Abstract

This is a sequel to [arXiv:1708.09092v2]. For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $\mathfrak{gl}(1|1)$-Alexander polynomial $\underline{\Delta }(G,c)$ defined by Viro satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092v2]. As a corollary we show that the Alexander polynomial ${\Delta }_{(G,c)}(t)$ studied in [arXiv:1708.09092v2] coincides with $\underline{\Delta }(G,c)$ for a positive coloring $c$ of $G$, where ${\Delta }_{(G,c)}(t)$ is constructed from certain regular covering space of the complement of $G$ in $S^3$ and it is the Euler characteristic of the Heegaard Floer homology of $G$ that we studied before. When $G$ is a plane graph, we provide a topological interpretation to the vertex state sum of $\underline{\Delta }(G,c)$ by considering a special Heegaard diagram of $G$ and the Fox calculus on the Heegaard surface.

26 pages. Some notations may vary slightly from published version

References in corpus (1)

• An Alexander polynomial for MOY graphs