{
  "id": "math/0506524",
  "version": "v4",
  "published": "2005-06-25T05:45:14.000Z",
  "updated": "2009-08-11T04:11:15.000Z",
  "title": "Topology of spaces of knots in dimension 3",
  "authors": [
    "Ryan Budney"
  ],
  "comment": "22 pages, 16 figures",
  "journal": "Proceedings of the London Mathematical Society Vol 101 pt. 2 (Sept. 2010)",
  "doi": "10.1112/plms/pdp058",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K, recursively, in terms of the homotopy types of `simpler' components of K, in the sense that they correspond to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot which has a hyperbolic manifold contained in the JSJ-decomposition of its complement.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-08-11T04:11:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R40",
      "57M25",
      "57M50",
      "57P48",
      "57R50"
    ],
    "keywords": [
      "homotopy type",
      "long knot",
      "simplest companionship trees",
      "shorter companionship trees",
      "isotopy class"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6524B"
    }
  }
}