{
  "id": "0802.2283",
  "version": "v2",
  "published": "2008-02-15T22:44:29.000Z",
  "updated": "2009-10-14T07:52:03.000Z",
  "title": "A refined Jones polynomial for symmetric unions",
  "authors": [
    "Michael Eisermann",
    "Christoph Lamm"
  ],
  "comment": "28 pages; v2: some improvements and corrections suggested by the referee",
  "categories": [
    "math.GT"
  ],
  "abstract": "Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_D(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_\\pm$: we obtain $W_D(t,t) = V_K(t)$ and $W_D(-1,t) = V_{K_-}(t) \\cdot V_{K_+}(t)$, which has the form of a symmetric product $f(t) \\cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-10-14T07:52:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "refined jones polynomial",
      "ribbon knot",
      "partial knots",
      "essentially distinct symmetric union presentations",
      "symmetric reidemeister moves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.2283E"
    }
  }
}