{
  "id": "math/0402025",
  "version": "v1",
  "published": "2004-02-03T03:35:20.000Z",
  "updated": "2004-02-03T03:35:20.000Z",
  "title": "On the growth rate of tunnel number of knots",
  "authors": [
    "Tsuyoshi Kobayashi",
    "Yo'av Rieck"
  ],
  "comment": "19 pages, 8 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Given a knot $K$ in a closed orientable manifold $M$ we define the growth rate of the tunnel number of $K$ to be $gr_t(K) = \\limsup_{n \\to \\infty} \\frac{t(nK) - n t(K)}{n-1}$. As our main result we prove that the Heegaard genus of $M$ is strictly less than the Heegaard genus of the knot exterior if and only if the growth rate is less than 1. In particular this shows that a non-trivial knot in $S^3$ is never asymptotically super additive. The main result gives conditions that imply falsehood of Morimoto's Conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-02-03T03:35:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M99"
    ],
    "keywords": [
      "growth rate",
      "tunnel number",
      "main result",
      "heegaard genus",
      "knot exterior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......2025K"
    }
  }
}