{
  "id": "math/0010212",
  "version": "v1",
  "published": "2000-10-22T21:13:40.000Z",
  "updated": "2000-10-22T21:13:40.000Z",
  "title": "Unknotting tunnels and Seifert surfaces",
  "authors": [
    "Martin Scharlemann",
    "Abigail Thompson"
  ],
  "comment": "29 pages, 20 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $K$ be a knot with an unknotting tunnel $\\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\\rho = p/q \\in \\mathbb{Q}/2 \\mathbb{Z}$, $p$ odd, defined for the pair $(K, \\gamma)$. The invariant $\\rho$ has interesting geometric properties: It is often straightforward to calculate; e. g. for $K$ a torus knot and $\\gamma$ an annulus-spanning arc, $\\rho(K, \\gamma) = 1$. Although $\\rho$ is defined abstractly, it is naturally revealed when $K \\cup \\gamma$ is put in thin position. If $\\rho \\neq 1$ then there is a minimal genus Seifert surface $F$ for $K$ such that the tunnel $\\gamma$ can be slid and isotoped to lie on $F$. One consequence: if $\\rho(K, \\gamma) \\neq 1$ then $genus(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \\gamma)$ with $\\rho(K, \\gamma) \\neq 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-10-22T21:13:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "unknotting tunnel",
      "minimal genus seifert surface",
      "thin position",
      "interesting geometric properties",
      "torus knot"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....10212S"
    }
  }
}