{
  "id": "math/0504305",
  "version": "v2",
  "published": "2005-04-14T19:19:03.000Z",
  "updated": "2009-04-30T12:07:45.000Z",
  "title": "The C-polynomial of a knot",
  "authors": [
    "Stavros Garoufalidis",
    "Xinyu Sun"
  ],
  "comment": "This is the version published by Algebraic & Geometric Topology on 11 October 2006",
  "journal": "Algebr. Geom. Topol. 6 (2006) 1623-1653",
  "doi": "10.2140/agt.2006.6.1623",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot. In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U_q(SL_2)) specializes at q=1 to the better known A-polynomial of a knot, which has to do with genuine SL_2(C) representations of the knot complement. Computing the non-commutative A-polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C-polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A-polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C-polynomial and the Alexander polynomial, all confirmed for the class of twist knots.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-04-30T12:07:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M25"
    ],
    "keywords": [
      "colored jones function",
      "non-commutative a-polynomial",
      "c-polynomial",
      "twist knots",
      "non-trivial linear q-difference equation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4305G"
    }
  }
}