{
  "id": "1610.07494",
  "version": "v1",
  "published": "2016-10-24T17:01:03.000Z",
  "updated": "2016-10-24T17:01:03.000Z",
  "title": "On a Heegaard Floer theory for tangles",
  "authors": [
    "Claudius Zibrowius"
  ],
  "comment": "PhD thesis (Cambridge, 2016, awaiting viva). This article supersedes arXiv:1601.04915. Comments welcome",
  "categories": [
    "math.GT",
    "math.QA",
    "math.SG"
  ],
  "abstract": "The purpose of this thesis is to define a \"local\" version of Ozsv\\'{a}th and Szab\\'{o}'s Heegaard Floer homology $\\operatorname{\\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology $\\operatorname{\\widehat{HFT}}$ for tangles in the closed 3-ball. After studying basic properties of $\\operatorname{\\widehat{HFT}}$ and its decategorified tangle invariant $\\nabla_T^s$, we prove a glueing theorem in terms of Zarev's bordered sutured Floer homology, which endows $\\operatorname{\\widehat{HFT}}$ with an additional glueing structure. For 4-ended tangles, we repackage this glueing structure into certain curved complexes $\\operatorname{CFT}^\\partial$, which we call peculiar modules. This allows us to easily recover oriented and unoriented skein relations for $\\operatorname{\\widehat{HFL}}$. Our peculiar modules enjoy some symmetry properties, which support a conjecture about $\\delta$-graded mutation invariance of $\\operatorname{\\widehat{HFL}}$. In fact, we show that any two links related by mutation about a $(2,-3)$-pretzel tangle have the same $\\delta$-graded link Floer homology. In the last part of this thesis, we explore the relationship between peculiar modules and twisted complexes in the fully wrapped Fukaya category of the 4-punctured sphere. This thesis is accompanied by two Mathematica packages. The first is a tool for computing the generators of $\\operatorname{\\widehat{HFT}}$ and its decategorified tangle invariant $\\nabla_T^s$. The second allows us to compute Zarev's bordered sutured Floer invariants of any bordered sutured manifold using nice diagrams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-24T17:01:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heegaard floer theory",
      "bordered sutured floer invariants",
      "heegaard floer homology",
      "peculiar modules",
      "decategorified tangle invariant"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}