{
  "id": "2101.04588",
  "version": "v1",
  "published": "2021-01-12T16:39:00.000Z",
  "updated": "2021-01-12T16:39:00.000Z",
  "title": "Distance $4$ curves on closed surfaces of arbitrary genus",
  "authors": [
    "Sreekrishna Palaparthi",
    "Kuwari Mahanta"
  ],
  "comment": "14 pages, 18 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $S_g$ denote a closed, orientable surface of genus $g \\geq 2$ and $\\mathcal{C}(S_g)$ the associated curve complex. The mapping class group of $S_g$ acts on $\\mathcal{C}(S_g)$ by isometries. Since Dehn twists about certain curves generate the mapping class group of $S_g$, one can ask how Dehn twists move specific vertices in $\\mathcal{C}(S_g)$ away from themselves. In this article, we answer this question for a specific case when the vertices are at a distance $3$. We show that if two curves represent vertices at a distance $3$ in $\\mathcal{C}(S_g)$ then the Dehn twist of one curve about another yields two vertices at distance $4$. This produces many examples of curves on $S_g$ which are at distance $4$ in $\\mathcal{C}(S_g)$. We also show that the minimum intersection number of any two curves at a distance $4$ on $S_g$, $i_{min}(g,4) \\leq (2g-1)^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-12T16:39:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M60",
      "20F65",
      "20F67"
    ],
    "keywords": [
      "arbitrary genus",
      "closed surfaces",
      "mapping class group",
      "dehn twists move specific vertices",
      "minimum intersection number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}