{
  "id": "1801.06301",
  "version": "v1",
  "published": "2018-01-19T05:31:17.000Z",
  "updated": "2018-01-19T05:31:17.000Z",
  "title": "A topological interpretation of Viro's $gl(1\\vert 1)$-Alexander polynomial of a graph",
  "authors": [
    "Yuanyuan Bao"
  ],
  "comment": "19 pages. Comments are welcome",
  "categories": [
    "math.GT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-19T05:31:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57M25"
    ],
    "keywords": [
      "alexander polynomial",
      "topological interpretation",
      "heegaard floer homology",
      "vertex state sum",
      "special heegaard diagram"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}