{
  "id": "1407.3321",
  "version": "v2",
  "published": "2014-07-11T23:19:51.000Z",
  "updated": "2014-11-12T20:19:09.000Z",
  "title": "Intersection numbers in the curve complex via subsurface projections",
  "authors": [
    "Yohsuke Watanabe"
  ],
  "comment": "25 pages. In v1, Theorem 1.3 was written for a general surface, since there was a mistake in the proof, now I have Theorem 1.4",
  "categories": [
    "math.GT"
  ],
  "abstract": "A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured in terms of their intersection number. In this paper, we show that the intersection number of two curves can be measured in terms of the sum of all large subsurface projection distances between them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-11T23:19:51.000Z",
      "title": "The intersection number of two curves and their subsurface projection distance",
      "abstract": "Suppose $\\xi(S)\\geq 1$. Let $x,y\\in C(S)$, we recall the following classical inequality which goes back to the work of Lickorish \\cite{LIC} and stated by Hempel \\cite{HEM} later on, $$d_{S}(x,y)\\prec \\log i(x,y).$$ In this paper, we improve on the inequality. We show that for sufficiently large $k$, $$\\log i(x,y) \\asymp \\sum_{Z}[ d_{Z}(x,y)]_{k}+\\sum_{A} \\log [d_{A}(x,y)]_{k}$$ where $Z$ ranges over all subsurfaces in $S$ which are not annuli, and $A$ ranges over all annuli in $S$.",
      "comment": "16 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-12T20:19:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "subsurface projection distance",
      "intersection number",
      "classical inequality",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3321W"
    }
  }
}