{
  "id": "1712.03486",
  "version": "v1",
  "published": "2017-12-10T08:26:59.000Z",
  "updated": "2017-12-10T08:26:59.000Z",
  "title": "Concordance invariants of doubled knots and blowing up",
  "authors": [
    "Se-Goo Kim",
    "Kwan Yong Lee"
  ],
  "comment": "7 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\nu$ be either the Ozsv\\'ath-Szab\\'o $\\tau$-invariant or half the Rasmussen $s$-invariant. For a knot $K$, Livingston and Naik defined the invariant $t_\\nu(K)$ to be the minimum of $t$ for which $\\nu$ of the $t$-twisted positive Whitehead double of $K$ vanishes. They proved that $t_\\nu(K)$ is bounded above by $-TB(-K)$, where $TB$ is the maximal Thurston-Bennequin number. We use blowing up process to find a crossing change formula and a new upper bound for $t_\\nu$ in terms of the unknotting number. As an application, we present infinitely many knots $K$ such that the difference between Livingston-Naik's upper bound $-TB(-K)$ and $t_\\nu(K)$ can be arbitrarily large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-10T08:26:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concordance invariants",
      "doubled knots",
      "livingston-naiks upper bound",
      "maximal thurston-bennequin number",
      "crossing change formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}