{
  "id": "2211.15290",
  "version": "v1",
  "published": "2022-11-28T13:24:49.000Z",
  "updated": "2022-11-28T13:24:49.000Z",
  "title": "Example of distance $5$ curves on closed surfaces",
  "authors": [
    "Kuwari Mahanta"
  ],
  "comment": "21 pages, 39 figures, 1 table",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $S_g$ denote a closed, orientable surface of genus $g \\geq 2$ and $\\mathcal{C}(S_g)$ be the associated curve graph. Let $d$ be the path metric on $\\mathcal{C}(S_g)$ and $a_0$ and $a_4$ be two curves on $S_g$ with $d(a_0, a_4) = 4$. It follows from the triangle inequality that $d(a_0, T_{a_4}(a_0)) \\leq 6$. In this article we give a criterion for when $d(a_0, T_{a_4}(a_0)) = 4$ and when $d(a_0, T_{a_4}(a_0)) \\geq 5$. We further give an explicit example of a pair of curves on $S_2$ which represent vertices at a distance $5$ in $\\mathcal{C}(S_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-28T13:24:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K20",
      "57M60"
    ],
    "keywords": [
      "closed surfaces",
      "path metric",
      "triangle inequality",
      "represent vertices",
      "associated curve graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}