A topological interpretation of Viro's $gl(1\vert 1)$-Alexander polynomial of a graph

Yuanyuan Bao

Published 2018-01-19Version 1

This is a sequel to our previous paper [arXiv:1708.09092]. For an oriented trivalent graph $G$ without source or sink embedded in $S^3$, we prove that the $gl(1\vert 1)$-Alexander polynomial $\underline{\Delta}(G, c)$ defined by Viro in [MR2255851] satisfies a series of relations, which we call MOY-type relations in [arXiv:1708.09092]. As a corollary we show that the Alexander polynomial $\Delta_{(G, c)}(t)$ studied in [arXiv:1708.09092] 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 $G$ studied in [arXiv:1401.6608v3]. When $G$ is a planar 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.