{
  "id": "2003.11909",
  "version": "v1",
  "published": "2020-03-25T01:40:33.000Z",
  "updated": "2020-03-25T01:40:33.000Z",
  "title": "Properties of minimal charts and their applications VI: the graph $Γ_{m+1}$ in a chart $Γ$ of type $(m;2,3,2)$",
  "authors": [
    "Teruo Nagase",
    "Akiko Shima"
  ],
  "comment": "39 pages, 31 figures. arXiv admin note: text overlap with arXiv:1902.00007, arXiv:1603.04639, arXiv:1609.08257",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\Gamma$ be a chart, and we denote by $\\Gamma_m$ the union of all the edges of label $m$. A chart $\\Gamma$ is of type $(m;2,3,2)$ if $w(\\Gamma)=7$, $w(\\Gamma_m\\cap\\Gamma_{m+1})=2$, $w(\\Gamma_{m+1}\\cap\\Gamma_{m+2})=3$, and $w(\\Gamma_{m+2}\\cap\\Gamma_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that if there is a minimal chart $\\Gamma$ of type $(m;2,3,2)$, then each of $\\Gamma_{m+1}$ and $\\Gamma_{m+2}$ contains one of three kinds of graphs. In the next paper, we shall prove that there is no minimal chart of type $(m;2,3,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-25T01:40:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q45",
      "57Q35"
    ],
    "keywords": [
      "minimal chart",
      "applications vi",
      "properties",
      "white vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}