{
  "id": "1306.4632",
  "version": "v2",
  "published": "2013-06-19T17:58:55.000Z",
  "updated": "2015-02-20T19:37:24.000Z",
  "title": "Satellite operators as group actions on knot concordance",
  "authors": [
    "Christopher W. Davis",
    "Arunima Ray"
  ],
  "comment": "20 pages, 9 figures; in the second version, we have added several new results about surjectivity of satellite operators, and inverses of satellite operators, and the exposition and structure of the paper have been improved",
  "categories": [
    "math.GT"
  ],
  "abstract": "Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\\overline{P}$ for which $\\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-19T17:58:55.000Z",
      "abstract": "Any knot in a solid torus (called a satellite operator) acts on knots in 3-space. We introduce a generalization of satellite operators which act on knots in homology 3-spheres. Unlike traditional satellite operators, these generalized operators form a group, modulo an appropriate generalization of concordance. By studying the action of this group on knots in homology 3-spheres we recover the very recent result of Cochran-Davis-Ray that satellite operators with strong winding number one give injective functions on smooth knot concordance in S^3 x [0,1], modulo the smooth 4-dimensional Poincare Conjecture. We also describe how the notion of generalized satellite operators provides a new framework within which to consider the question of surjectivity of satellite operators and make some progress towards answering this question. We also construct a new example of a bijective satellite operator.",
      "comment": "19 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-20T19:37:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "group actions",
      "unlike traditional satellite operators",
      "smooth knot concordance",
      "poincare conjecture",
      "solid torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.4632D"
    }
  }
}