LAUSR

Abstract For an oriented trivalent graph G without source or sink embedded in S 3 , 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 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 Δ _ ( G , c ) by considering a special Heegaard diagram of G and the Fox calculus on the Heegaard surface.