{
  "id": "1405.0143",
  "version": "v2",
  "published": "2014-05-01T12:25:10.000Z",
  "updated": "2014-09-24T09:29:41.000Z",
  "title": "An infinite family of prime knots with a certain property for the clasp number",
  "authors": [
    "Teruhisa Kadokami",
    "Kengo Kawamura"
  ],
  "comment": "13 pages, 12 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "The clasp number $c(K)$ of a knot $K$ is the minimum number of clasp singularities among all clasp disks bounded by $K$. It is known that the genus $g(K)$ and the unknotting number $u(K)$ are lower bounds of the clasp number, that is, $\\max\\{g(K),u(K)\\} \\leq c(K)$. Then it is natural to ask whether there exists a knot $K$ such that $\\max\\{g(K),u(K)\\}<c(K)$. In this paper, we prove that there exists an infinite family of prime knots such that the question above is affirmative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-01T12:25:10.000Z",
      "title": "Tabulation of the clasp numbers of prime knots with up to $10$ crossings",
      "abstract": "Any knot in $S^{3}$ bounds a clasp disk that is an immersed disk in $S^{3}$ whose singular set consists of only clasps. The clasp number of a knot $K$, denoted by $c(K)$, is the minimal number of clasps among all clasp disks bounded by $K$. We determine $c(10_{97})=3$ by applying a characterization of the Alexander module via a clasp disk due to K. Morimoto. Moreover we investigate the clasp numbers of $2$-bridge knots, and show a table for the clasp numbers of prime knots with up to $10$ crossings.",
      "comment": "11 pages, 8 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-24T09:29:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "clasp number",
      "prime knots",
      "tabulation",
      "singular set consists",
      "alexander module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0143K"
    }
  }
}