{
  "id": "math/0301320",
  "version": "v1",
  "published": "2003-01-28T00:35:16.000Z",
  "updated": "2003-01-28T00:35:16.000Z",
  "title": "On the bridge number of knot diagrams with minimal crossings",
  "authors": [
    "Jae-Wook Chung",
    "Xiao-Song Lin"
  ],
  "comment": "18 pages, 7 figures",
  "doi": "10.1017/S0305004104007753",
  "categories": [
    "math.GT"
  ],
  "abstract": "Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\\sqrt{1+c(D)} \\leq b(D)\\leq c(D)$. These inequalities are shape in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-01-28T00:35:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "knot diagrams",
      "minimal crossings",
      "bridge number",
      "torus knot",
      "alternating knot"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2004,
      "month": "Nov",
      "volume": 137,
      "number": 3,
      "pages": 617
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004MPCPS.137..617C"
    }
  }
}