{
  "id": "1603.00846",
  "version": "v1",
  "published": "2016-03-02T19:56:58.000Z",
  "updated": "2016-03-02T19:56:58.000Z",
  "title": "Mapping class group orbits of curves with self-intersections",
  "authors": [
    "Patricia Cahn",
    "Federica Fanoni",
    "Bram Petri"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.CO",
    "math.DG"
  ],
  "abstract": "We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of number of such orbits, on a genus g surface with k self-intersections, for k bounded and $g\\to\\infty$. To do this we count embeddings of ribbon graphs. As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the genus of the surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-02T19:56:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-intersections",
      "study mapping class group orbits",
      "contain short curves",
      "isotopy classes",
      "ribbon graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160300846C"
    }
  }
}