{
  "id": "1704.01232",
  "version": "v1",
  "published": "2017-04-05T01:05:35.000Z",
  "updated": "2017-04-05T01:05:35.000Z",
  "title": "The structure of a minimal $n$-chart with two crossings I: Complementary domains of $Γ_1\\cupΓ_{n-1}$",
  "authors": [
    "Teruo Nagase",
    "Akiko Shima"
  ],
  "comment": "40 pages, 21 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label $m$ of a chart $\\Gamma$ we denote by $\\Gamma_m$ the union of all the edges of label $m$ and their vertices. For a minimal chart $\\Gamma$ with exactly two crossings, we can show that the two crossings are contained in $\\Gamma_\\alpha\\cap\\Gamma_\\beta$ for some labels $\\alpha<\\beta$. In this paper, we study the structure of a disk $D$ not containing any crossing but satisfying $\\Gamma\\cap \\partial D\\subset\\Gamma_{\\alpha+1}\\cup \\Gamma_{\\beta-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-05T01:05:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q45",
      "57Q35"
    ],
    "keywords": [
      "complementary domains",
      "minimal chart",
      "first step"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}