{
  "id": "math/0406106",
  "version": "v4",
  "published": "2004-06-06T22:23:09.000Z",
  "updated": "2008-10-23T12:23:15.000Z",
  "title": "On Knot Polynomials of Annular Surfaces and their Boundary Links",
  "authors": [
    "Hermann Gruber"
  ],
  "comment": "Version 4: revision as of October 10, 2008. Fixed several errors and inaccuracies. 11 pages, 1 figure. To appear in Mathematical Proceedings of the Cambridge Philosophical Society",
  "journal": "Mathematical Proceedings of the Cambridge Philosophical Society 147 (2009) 173-183",
  "doi": "10.1017/S0305004109002370",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "Stoimenow and Kidwell asked the following question: Let $K$ be a non-trivial knot, and let $W(K)$ be a Whitehead double of $K$. Let $F(a,z)$ be the Kauffman polynomial and $P(v,z)$ the skein polynomial. Is then always $\\max\\deg_z P_{W(K)} - 1 = 2 \\max\\deg_z F_K$? Here this question is rephrased in more general terms as a conjectured relation between the maximum $z$-degrees of the Kauffman polynomial of an annular surface $A$ on the one hand, and the Rudolph polynomial on the other hand, the latter being defined as a certain M\\\"obius transform of the skein polynomial of the boundary link $\\partial A$. That relation is shown to hold for algebraic alternating links, thus simultaneously solving the conjecture by Kidwell and Stoimenow and a related conjecture by Tripp for this class of links. Also, in spite of the heavyweight definition of the Rudolph polynomial $\\{K\\}$ of a link $K$, the remarkably simple formula $\\{\\bigcirc\\}\\{L#M\\}=\\{L\\}\\{M\\}$ for link composition is established. This last result can be used to reduce the conjecture in question to the case of prime links.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-10-23T12:23:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57M25"
    ],
    "keywords": [
      "boundary link",
      "annular surface",
      "knot polynomials",
      "skein polynomial",
      "kauffman polynomial"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6106G"
    }
  }
}