{
  "id": "1711.11290",
  "version": "v1",
  "published": "2017-11-30T09:48:26.000Z",
  "updated": "2017-11-30T09:48:26.000Z",
  "title": "Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot",
  "authors": [
    "Ka Ho Wong",
    "Thomas Kwok-Keung Au"
  ],
  "comment": "43 pages, 0 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we investigate the asymptotic behavior of the colored Jones polynomial and the Turaev-Viro invariant for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomial evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. Generalizing the work of \\cite{WA17} and \\cite{HM13}, we obtain the asymptotic expansion formula (AEF) of the colored Jones polynomial of figure eight knot with $s$ close to $1$. An upper bound for the asymptotic expansion formula of the colored Jones polynomial of figure eight knot with $s$ close to $1/2$ is also obtained. From the result in \\cite{DKY17}, the Turaev Viro invariant of figure eight knot can be expressed in terms of a sum of its colored Jones polynomials. Our results show that this sum is asymptotically equal to the sum of the terms with $s$ close to 1. As an application of the asymptotic behavior of the colored Jones polynomials, we obtain the asymptotic expansion formula for the Turaev-Viro invariant of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic knots.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-30T09:48:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57M50"
    ],
    "keywords": [
      "turaev-viro invariant",
      "asymptotic behavior",
      "asymptotic expansion formula",
      "th colored jones polynomial",
      "general hyperbolic knots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}