{
  "id": "math/0306230",
  "version": "v4",
  "published": "2003-06-15T17:12:51.000Z",
  "updated": "2004-09-20T21:43:10.000Z",
  "title": "On the characteristic and deformation varieties of a knot",
  "authors": [
    "Stavros Garoufalidis"
  ],
  "comment": "Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper12.abs.html",
  "journal": "Geom. Topol. Monogr. 7 (2004) 291-309",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q-holonomic, ie, that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1-dimensional variety in C^2. We then compare it with the character variety of SL_2(C) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots. We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the so-called noncommutative A-polynomial) of the characteristic variety of a knot. Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2004-09-20T21:43:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M25"
    ],
    "keywords": [
      "colored jones function",
      "deformation varieties",
      "nontrivial linear recursion relation",
      "characteristic variety",
      "higher rank groups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......6230G"
    }
  }
}