{
  "id": "2005.01576",
  "version": "v1",
  "published": "2020-05-04T15:43:55.000Z",
  "updated": "2020-05-04T15:43:55.000Z",
  "title": "Group Presentations for Links in Thickened Surfaces",
  "authors": [
    "Daniel S. Silver",
    "Susan G. Williams"
  ],
  "comment": "16 pages, 12 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links $\\ell$ in a thickened surface $S \\times [0,1]$. Their precise relationship, as given in the 2012 thesis of R.E. Byrd, is established here by an elementary argument. When a diagram in $S$ for $\\ell$ can be checkerboard shaded, the Dehn presentation leads naturally to an abelian \"Dehn coloring group,\" an isotopy invariant of $\\ell$. Introducing homological information from $S$ produces a stronger invariant, $\\cal C$, a module over the group ring of $H_1(S; {\\mathbb Z})$. The authors previously defined the Laplacian modules ${\\cal L}_G,{ \\cal L}_{G^*}$ and polynomials $\\Delta_G, \\Delta_{G^*}$ associated to a Tait graph $G$ and its dual $G^*$, and showed that the pairs $\\{{\\cal L}_G, {\\cal L}_{G^*}\\}$, $\\{\\Delta_G, \\Delta_{G^*}\\}$ are isotopy invariants of $\\ell$. The relationship between $\\cal C$ and the Laplacian modules is described and used to prove that $\\Delta_G$ and $\\Delta_{G^*}$ are equal when $S$ is a torus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-04T15:43:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "05C10"
    ],
    "keywords": [
      "thickened surface",
      "group presentations",
      "isotopy invariant",
      "dehn presentation",
      "laplacian modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}