{
  "id": "2307.07100",
  "version": "v1",
  "published": "2023-07-14T00:15:32.000Z",
  "updated": "2023-07-14T00:15:32.000Z",
  "title": "The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation",
  "authors": [
    "Hitoshi Murakami"
  ],
  "comment": "58 pages, 21 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\\exp\\bigl((\\kappa+2p\\pi\\i/N\\bigr)$, where $\\kappa:=\\arccosh(3/2)$ and $p$ is a positive integer. We can prove that it grows exponentially with growth rate determined by the Chern--Simons invariant of an affine representation from the fundamental group of the knot complement to the Lie group $\\SL(2;\\C)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T00:15:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "affine representation",
      "figure eight-knot",
      "dimensional colored jones polynomial",
      "knot complement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}