{
  "id": "0909.1162",
  "version": "v3",
  "published": "2009-09-07T08:26:32.000Z",
  "updated": "2010-11-17T02:48:15.000Z",
  "title": "Bridge position and the representativity of spatial graphs",
  "authors": [
    "Makoto Ozawa"
  ],
  "comment": "16 pages, 9 figures. In version 2, Theorem 4.3 (in version 3) was added. In version 3, Theorem 1.6 was added",
  "categories": [
    "math.GT"
  ],
  "abstract": "First, we extend Otal's result for the trivial knot to trivial spatial graphs, namely, we show that for any bridge tangle decomposing sphere $S^2$ for a trivial spatial graph $\\Gamma$, there exists a 2-sphere $F$ such that $F$ contains $\\Gamma$ and $F$ intersects $S^2$ in a single loop. Next, we introduce two invariants for spatial graphs. As a generalization of the bridge number for knots, we define the {\\em bridge string number} $bs(\\Gamma)$ of a spatial graph $\\Gamma$ as the minimal number of $|\\Gamma\\cap S^2|$ for all bridge tangle decomposing sphere $S^2$. As a spatial version of the representativity for a graph embedded in a surface, we define the {\\em representativity} of a non-trivial spatial graph $\\Gamma$ as \\[ r(\\Gamma)=\\max_{F\\in\\mathcal{F}} \\min_{D\\in\\mathcal{D}_F} |\\partial D\\cap \\Gamma|, \\] where $\\mathcal{F}$ is the set of all closed surfaces containing $\\Gamma$ and $\\mathcal{D}_F$ is the set of all compressing disks for $F$ in $S^3$. Then we show that for a non-trivial spatial graph $\\Gamma$, \\[ \\displaystyle r(\\Gamma)\\le \\frac{bs(\\Gamma)}{2}. \\] In particular, if $\\Gamma$ is a knot, then $r(\\Gamma)\\le b(\\Gamma)$, where $b(\\Gamma)$ denotes the bridge number. This generalizes Schubert's result on torus knots.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-11-17T02:48:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57Q35"
    ],
    "keywords": [
      "bridge position",
      "bridge tangle decomposing sphere",
      "representativity",
      "non-trivial spatial graph",
      "bridge number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1162O"
    }
  }
}