{
  "id": "math/0605314",
  "version": "v1",
  "published": "2006-05-11T19:18:15.000Z",
  "updated": "2006-05-11T19:18:15.000Z",
  "title": "A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres",
  "authors": [
    "Kazuo Habiro"
  ],
  "comment": "66 pages, 8 figures",
  "doi": "10.1007/s00222-007-0071-0",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "We construct an invariant J_M of integral homology spheres M with values in a completion \\hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity \\zeta gives the the SU(2) Witten-Reshetikhin-Turaev invariant \\tau_\\zeta(M) of M at \\zeta. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, \\tau_\\zeta(M) is an algebraic integer. Moreover, it follows that \\tau_\\zeta(M) as a function on \\zeta behaves like an ``analytic function'' defined on the set of roots of unity. That is, the \\tau_\\zeta(M) for all roots of unity are determined by a \"Taylor expansion\" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, \\tau_\\zeta(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q=1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-05-11T19:18:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "17B37"
    ],
    "keywords": [
      "integral homology spheres",
      "unified witten-reshetikhin-turaev invariant",
      "taylor expansion",
      "prime power orders",
      "algebraic integer"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Inventiones Mathematicae",
      "year": 2007,
      "month": "Sep",
      "volume": 171,
      "number": 1,
      "pages": 1
    },
    "note": {
      "typesetting": "TeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007InMat.171....1H"
    }
  }
}