{
  "id": "1206.5989",
  "version": "v2",
  "published": "2012-06-26T13:51:26.000Z",
  "updated": "2015-10-07T17:35:46.000Z",
  "title": "Localization and the link Floer homology of doubly-periodic knots",
  "authors": [
    "Kristen Hendricks"
  ],
  "comment": "This is the final version published in the Journal of Symplectic Geometry. Typos corrected, introduction reorganized, Corollary 1.5 expanded, and Example 1.9 added. Title updated from former preprint title: \"A note on the link Floer homology of doubly-periodic knots.\"",
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "A knot \\widetilde{K} \\subset S^3 is q-periodic if there is a \\mathbb Z_q-action preserving \\widetilde{K} whose fixed set is an unknot U. The quotient of \\widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For \\widetilde{K} a two-periodic knot, we show there is a spectral sequence whose E^1 page is \\hat{\\mathit{HFL}}(S^3,\\widetilde{K}\\cup U)\\otimes V^{\\otimes (2n-1)})\\otimes \\mathbb Z_2((\\theta)) and whose E^{\\infty} pages is isomorphic to (\\hat{\\mathit{HFL}}(S^3,K\\cup U)\\otimes V^{\\otimes (n-1)})\\otimes \\mathbb Z_2((\\theta)), as \\mathbb Z_2((\\theta))-modules, and a related spectral sequence whose E^1 page is (\\hat{\\mathit{HFK}}(S^3,\\widetilde{K})\\otimes V^{\\otimes (2n-1)}\\otimes W)\\otimes \\mathbb Z_2((\\theta)) and whose E^{\\infty} page is isomorphic to (\\hat{\\mathit{HFK}}(S^3,K)\\otimes V^{\\otimes (n-1)} \\otimes W)\\otimes \\mathbb Z_2((\\theta)). As a consequence, we use these spectral sequences to recover a classical lower bound of Edmonds on the genus of \\widetilde{K}, along with a weak version of a classical fibredness result of Edmonds and Livingston.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-26T13:51:26.000Z",
      "title": "A note on the link Floer homology of doubly-periodic knots",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-07T17:35:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27",
      "57R58",
      "53D40"
    ],
    "keywords": [
      "link floer homology",
      "doubly-periodic knots",
      "spectral sequence",
      "construct equivariant heegaard diagrams",
      "q-periodic knots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.5989H"
    }
  }
}