{
  "id": "2008.10168",
  "version": "v1",
  "published": "2020-08-24T02:53:21.000Z",
  "updated": "2020-08-24T02:53:21.000Z",
  "title": "Quivers with potentials associated to triangulations of closed surfaces with at most two punctures",
  "authors": [
    "Jan Geuenich",
    "Daniel Labardini-Fragoso",
    "José Luis Miranda-Olvera"
  ],
  "comment": "19 pages, 5 figures",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We tackle the classification problem of non-degenerate potentials for quivers arising from triangulations of surfaces in the cases left open by Geiss-Labardini-Schr\\\"oer. Namely, for once-punctured closed surfaces of positive genus, we show that the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not weakly right-equivalent; we do so by showing that the potentials obtained by adding the 3-cycles coming from triangles and a fixed power of the cycle surrounding the puncture are well behaved under flips and QP-mutations. For twice-punctured closed surfaces of positive genus, we prove that the quiver of any triangulation admits exactly one non-degenerate potential up to weak right-equivalence, thus confirming the veracity of a conjecture of the aforementioned authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-24T02:53:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16P10",
      "16G20",
      "13F60",
      "57N05",
      "05E99"
    ],
    "keywords": [
      "non-degenerate potential",
      "triangulation admits",
      "positive genus",
      "cases left open",
      "classification problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}