{
  "id": "2010.13162",
  "version": "v1",
  "published": "2020-10-25T17:01:06.000Z",
  "updated": "2020-10-25T17:01:06.000Z",
  "title": "4-dimensional aspects of tight contact 3-manifolds",
  "authors": [
    "Matthew Hedden",
    "Katherine Raoux"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: if a fibered link L induces a tight contact structure on Y then its fiber surface maximize Euler characteristic amongst all surfaces in Yx[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsv\\'ath-Szab\\'o contact invariant. We also show that any subsurface of a page of an open book inducing a contact structure with non-trivial invariant maximize \"slice\" Euler-characteristic for its boundary, and conjecture that this holds more generally for open books inducing tight contact structures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-25T17:01:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K18",
      "57K10",
      "57K41",
      "57K31",
      "57R58",
      "57R65",
      "57K33",
      "53D10",
      "53D40",
      "57K33",
      "57K40",
      "57K43"
    ],
    "keywords": [
      "conjecture",
      "open books inducing tight contact",
      "books inducing tight contact structures",
      "fiber surface maximize euler characteristic",
      "non-vanishing ozsvath-szabo contact invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}