{
  "id": "math/0508100",
  "version": "v6",
  "published": "2005-08-04T20:07:39.000Z",
  "updated": "2011-08-31T18:25:46.000Z",
  "title": "Asymptotics of the colored Jones function of a knot",
  "authors": [
    "Stavros Garoufalidis",
    "Thang T. Q. Le"
  ],
  "comment": "31 pages, 13 figures",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the $n$th colored Jones polynomial at $e^{\\a/n}$, when $\\a$ is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1/n$ when $\\a$ is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the $n$th colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when $\\a$ is near $2 \\pi i$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2011-08-31T18:25:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M25"
    ],
    "keywords": [
      "colored jones function",
      "th colored jones polynomial",
      "generalized volume conjecture",
      "asymptotic behavior",
      "upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8100G"
    }
  }
}