{
  "id": "math/0501394",
  "version": "v2",
  "published": "2005-01-23T18:40:45.000Z",
  "updated": "2005-10-21T14:38:07.000Z",
  "title": "The Alexander polynomial of (1,1)-knots",
  "authors": [
    "Alessia Cattabriga"
  ],
  "comment": "11 pages, 1 figure. A corollary has been extended, and a new example added. Accepted for publication on J. Knot Theory Ramifications",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot, which we call the n-cyclic polynomial. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S^2\\times S^1, a result obtained by J. Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in the 3-sphere. As corollaries some properties of the Alexander polynomial of knots in the 3-sphere are extended to the case of (1,1)-knots in lens spaces.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-10-21T14:38:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M12",
      "57M27"
    ],
    "keywords": [
      "alexander polynomial",
      "cyclic presentation",
      "fundamental group",
      "n-cyclic polynomial",
      "lens spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1394C"
    }
  }
}