{
  "id": "math/9909050",
  "version": "v4",
  "published": "1999-09-09T12:03:27.000Z",
  "updated": "2001-12-20T16:55:19.000Z",
  "title": "Vassiliev invariants and rational knots of unknotting number one",
  "authors": [
    "A. Stoimenow"
  ],
  "comment": "13 pages, 2 figures; revision 26 Nov 99: added reference [OTY], discussion of signatures, branched cover homology and 4-genera, more problems; revision 7 Sep 01: Theorem 1.2 slightly improved, a few other minor structural changes; revision 20 Dec 01: final version, Theorem 1.2 improved, 2 sections removed",
  "journal": "Topology 42(1) (2003), 227--241.",
  "categories": [
    "math.GT"
  ],
  "abstract": "Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and Yamada). The same result is shown for 4-genera and finite reductions of the homology group of the double branched cover. Closer consideration is given to rational knots, where it is shown that the number of $n$-trivial rational knots of at most $k$ crossings is for any $n$ asymptotically at least $C^{(\\ln k)^2}$ for any $C<\\sqrt[2\\ln 2]{e}$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2001-12-20T16:55:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "vassiliev invariants",
      "trivial rational tangles",
      "trivial rational knots",
      "arbitrary unknotting number",
      "finite reductions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......9050S"
    }
  }
}