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

摘要

For an oriented trivalent graph G without source or sink embedded in S-3, we prove that the gl(1 vertical bar 1)-Alexander polynomial (Delta) under bar (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 Delta((G,c)) (t) studied in [3] coincides with (Delta) under bar (G, c) for a positive coloring c of G, where Delta((G,c)) 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 (Delta) under bar (G, c) by considering a special Heegaard diagram of G and the Fox calculus on the Heegaard surface. (C) 2019 Elsevier B.V. All rights reserved.