{
  "id": "math/0607258",
  "version": "v5",
  "published": "2006-07-11T13:29:23.000Z",
  "updated": "2012-02-19T15:37:10.000Z",
  "title": "Behavior of knot invariants under genus 2 mutation",
  "authors": [
    "Nathan M. Dunfield",
    "Stavros Garoufalidis",
    "Alexander Shumakovitch",
    "Morwen Thistlethwaite"
  ],
  "comment": "24 pages, 7 figures. v2: Added references. v3: Final published version; v4: Includes erratum correcting proofs of results depending on Proposition 2.7, which is false",
  "journal": "New York J. Math., 16 (2010) 99-123",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can be achieved by a sequence of at most two genus 2 mutations, the invariants that are preserved by genus 2 mutation are a proper subset of those preserved by Conway mutation. In particular, while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT polynomial is not. In the case of the sl_2-Khovanov homology, which may or may not be invariant under Conway mutation, we give an example where genus 2 mutation changes this homology. Finally, using these techniques, we exhibit examples of knots with the same same colored Jones polynomials, HOMFLY-PT polynomial, Kauffman polynomial, signature and volume, but different Khovanov homology.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2012-02-19T15:37:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10"
    ],
    "keywords": [
      "knot invariants",
      "homfly-pt polynomial",
      "better-known conway mutation",
      "paper compares genus",
      "mutation changes"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7258D"
    }
  }
}