{
  "id": "2008.04836",
  "version": "v1",
  "published": "2020-08-11T16:32:49.000Z",
  "updated": "2020-08-11T16:32:49.000Z",
  "title": "A polynomial invariant for veering triangulations",
  "authors": [
    "Michael Landry",
    "Yair N. Minsky",
    "Samuel J. Taylor"
  ],
  "comment": "50 pages, 15 figures",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "We introduce a polynomial invariant $V_\\tau \\in \\mathbb{Z}[H_1(M)/\\text{torsion}]$ associated to a veering triangulation $\\tau$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_\\tau$ recovers the Teichm\\\"uller polynomial of the fibered faces canonically associated to $\\tau$. Via Dehn filling, this gives a combinatorial description of the Teichm\\\"uller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $\\tau$, we show that the surfaces carried by $\\tau$ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is $\\textit{equal}$ to the cone over a (generally non-fibered) face of the Thurston norm ball, and that $\\tau$ computes the norm on this cone in a precise sense. We also give a combinatorial description of $V_\\tau$ in terms of the $\\textit{flow graph}$ for $\\tau$ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm determined by $\\tau$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-11T16:32:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial invariant",
      "combinatorial description",
      "thurston norm ball",
      "precise sense",
      "veering triangulation comes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}