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

TL;DR

This paper provides a topological interpretation of Viro's $rak{gl}(1|1)$-Alexander polynomial for graphs, establishing relations and equivalences with previously studied invariants, and offers a geometric perspective for plane graphs.

Contribution

It proves that Viro's $rak{gl}(1|1)$-Alexander polynomial satisfies MOY-type relations and relates it to the Euler characteristic of Heegaard Floer homology, with a topological interpretation for plane graphs.

Findings

$rak{gl}(1|1)$-Alexander polynomial satisfies MOY-type relations.

The polynomial coincides with a previously studied Alexander polynomial for positive colorings.

A topological interpretation for plane graphs using Heegaard diagrams and Fox calculus.

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.