A topological interpretation of Viro's [formula omitted]-Alexander polynomial of a graph

Abstract

For an oriented trivalent graph G without source or sink embedded in S3, we prove that the gl(1|1)-Alexander polynomial Δ_(G,c) defined by Viro satisfies a series of relations, which we call MOY-type relations in [3]. As a corollary we show that the Alexander polynomial Δ(G,c)(t) studied in [3] coincides with Δ_(G,c) for a positive coloring c of G, where Δ(G,c)(t) is constructed from a certain regular covering space of the complement of G in S3 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 Δ_(G,c) by considering a special Heegaard diagram of G and the Fox calculus on the Heegaard surface.