{
  "id": "math/0506523",
  "version": "v5",
  "published": "2005-06-25T05:26:30.000Z",
  "updated": "2007-10-29T07:54:00.000Z",
  "title": "JSJ-decompositions of knot and link complements in the 3-sphere",
  "authors": [
    "Ryan Budney"
  ],
  "comment": "30 pages, 15 figures. V5: minor revision. Notes for V5: Fixed a typo in the statement of the Brunnian properties, and clarified a counter-intuitive convention that knots are considered `split' for the purposes of a theorem of Kanenobu",
  "journal": "L'enseignement Mathe'matique (2) 52 (2006), 319--359",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in the 3-sphere. Formulated in the language of graphs, the result is the construction of a bijective correspondence between the isotopy classes of links in $S^3$ and a class of vertex-labelled, finite acyclic graphs, called companionship graphs. This construction can be thought of as a uniqueness theorem for Schubert's `satellite operations.' We identify precisely which graphs are companionship graphs of knots and links respectively. We also describe how a large family of operations on knots and links affects companionship graphs. This family of operations is called `splicing' and includes, among others, the operations of: cabling, connect-sum, Whitehead doubling and the deletion of a component.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2007-10-29T07:54:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M50",
      "57M15"
    ],
    "keywords": [
      "link complements",
      "jsj-decomposition",
      "links affects companionship graphs",
      "finite acyclic graphs",
      "uniqueness theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6523B"
    }
  }
}