{
  "id": "0802.2874",
  "version": "v3",
  "published": "2008-02-20T15:02:05.000Z",
  "updated": "2015-04-22T07:31:16.000Z",
  "title": "Floer homology and splicing knot complements",
  "authors": [
    "Eaman Eftekhary"
  ],
  "comment": "Some errors in version 2 of the paper are corrected, and the exposition is slightly improved",
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $\\widehat{\\mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $\\widehat{\\mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$\\big|(h_\\infty^1-h_1^1)(h_\\infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)\\big|.$$ We also show that if splicing the complement of a knot $K\\subset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-05T06:07:38.000Z",
      "abstract": "We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $\\widehat{\\mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $\\widehat{\\mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$\\big|(h_\\infty^1-h_1^1)(h_\\infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)\\big|.$$",
      "comment": "A major mistake in the splicing formula presented in the original version of the paper is fixed. Some computations are added. The argument for the results of the preprint arXiv:0807.2326 which were built on the splicing formula of the initial version of this paper no longer goes through",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-22T07:31:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57R58"
    ],
    "keywords": [
      "splicing knot complements",
      "heegaard floer homology",
      "knot floer homology",
      "heegaard floer group",
      "hat theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.2874E"
    }
  }
}