{
  "id": "1407.5987",
  "version": "v1",
  "published": "2014-07-22T19:49:55.000Z",
  "updated": "2014-07-22T19:49:55.000Z",
  "title": "Mirror links have dual odd and generalized Khovanov homology",
  "authors": [
    "Wojciech Lubawski",
    "Krzysztof K. Putyra"
  ],
  "comment": "18 pages",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "We show that the generalized Khovanov homology, defined by the second author in the framework of chronological cobordisms, admits a grading by the group $\\mathbb{Z}\\times\\mathbb{Z}_2$, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring $\\mathbb{Z}_{\\pi}:=\\mathbb{Z}[\\pi]/(\\pi^2-1)$ (here, setting $\\pi$ to $\\pm 1$ results either in even or odd Khovanov homology). The generalized homology has $\\Bbbk := \\mathbb{Z}[X,Y,Z^{\\pm 1}]/(X^2=Y^2=1)$ as coefficients, and the above implies that most of automorphisms of $\\Bbbk$ fix the isomorphism class of the generalized homology regarded as $\\Bbbk$-modules, so that the even and odd Khovanov homology are the only two specializations of the invariant. In particular, switching $X$ with $Y$ induces a derived isomorphism between the generalized Khovanov homology of a link $L$ with its dual version, i.e. the homology of the mirror image $L^!$, and we compute an explicit formula for this map. When specialized to integers it descends to a duality isomorphism for odd Khovanov homology, which was conjectured by A. Shumakovitch.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-22T19:49:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N35",
      "57M27"
    ],
    "keywords": [
      "generalized khovanov homology",
      "odd khovanov homology",
      "dual odd",
      "mirror links",
      "generalized homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5987L"
    }
  }
}