{
  "id": "0901.2974",
  "version": "v1",
  "published": "2009-01-20T03:01:35.000Z",
  "updated": "2009-01-20T03:01:35.000Z",
  "title": "Self-intersection numbers of curves on the punctured torus",
  "authors": [
    "Moira Chas",
    "Anthony Phillips"
  ],
  "comment": "39 pages, 8 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L-2)^2/4 if L is even, and (L-1)(L-3)/4 if L is odd. The classes attaining this bound are explicitly described in terms of the generators; there are (L-2)^2 + 4 of them if L is even, and 2(L-1)(L-3) + 8 if L is odd; similar descriptions and totals are given for classes with self-intersection number equal to one less than the maximum. Proofs use both combinatorial calculations and topological operations on representative curves. Computer-generated data are tabulated counting, for each non-negative integer, how many length-L classes have that self-intersection number, for each length L less than or equal to 12. Experimental data are also presented for the pair-of-pants surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-01-20T03:01:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "57N50"
    ],
    "keywords": [
      "punctured torus",
      "self-intersection number equal",
      "free homotopy class",
      "pair-of-pants surface",
      "combinatorial calculations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}