{
  "id": "1001.4568",
  "version": "v1",
  "published": "2010-01-25T23:45:05.000Z",
  "updated": "2010-01-25T23:45:05.000Z",
  "title": "Self-intersection numbers of curves in the doubly-punctured plane",
  "authors": [
    "Moira Chas",
    "Anthony Phillips"
  ],
  "comment": "15 pages, 8 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of the standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by L^2/4 + L/2 - 1, and that when L is even, this bound is sharp; in that case there are exactly four distinct classes attaining that bound. When L is odd, we establish a smaller, conjectured upper bound ((L^2 - 1)/4)) in certain cases; and there we show it is sharp. Furthermore, for the doubly-punctured plane, these self-intersection numbers are bounded below, by L/2 - 1 if L is even, (L - 1)/2 if L is odd; these bounds are sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-25T23:45:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "57N50"
    ],
    "keywords": [
      "self-intersection number",
      "doubly-punctured plane",
      "free homotopy class",
      "minimum number",
      "self-intersection points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1001.4568C"
    }
  }
}