{
  "id": "1309.7532",
  "version": "v2",
  "published": "2013-09-29T04:22:00.000Z",
  "updated": "2014-08-28T21:48:20.000Z",
  "title": "Casson towers and filtrations of the smooth knot concordance group",
  "authors": [
    "Arunima Ray"
  ],
  "comment": "30 pages, 21 figures; version 2 has 34 pages and 22 figures, more detailed discussion and better exposition at several places due to comments from an anonymous referee, added the word `smooth' in the title, to appear in Algebraic & Geometric Topology",
  "categories": [
    "math.GT"
  ],
  "abstract": "The n-solvable filtration $\\{\\mathcal{F}_n\\}_{n=0}^\\infty$ of the smooth knot concordance group (denoted by $\\mathcal{C}$), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\\mathcal{C}$ due to Cochran-Harvey-Horn, the positive and negative filtrations, denoted by $\\{\\mathcal{P}_n\\}_{n=0}^\\infty$ and $\\{\\mathcal{N}_n\\}_{n=0}^\\infty$ respectively. In particular, we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball with only positive (resp. negative) kinks in the base-level kinky disk, then K is in $\\mathcal{P}_n$ (resp. $\\mathcal{N}_n$). En route to this result we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and is therefore, n-solvable (this also implies that topologically slice knots bound arbitrarily tall gropes in the 4-ball). We also define a variant of Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it is n-solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K is in $\\mathcal{P}_n$ (resp. $\\mathcal{N}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot which is not topologically slice but lies in each $\\mathcal{F}_n$. We also give a 3-dimensional characterization, up to concordance, of knots which bound kinky disks in the 4-ball with only positive (resp. negative) kinks; such knots form a subset of $\\mathcal{P}_0$ (resp. $\\mathcal{N}_0$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-29T04:22:00.000Z",
      "title": "Casson towers and filtrations of the knot concordance group",
      "abstract": "The n-solvable filtration {F_n}_{n=0}^\\infty of the smooth knot concordance group (denoted by C), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of C due to Cochran-Harvey-Horn, the positive and negative filtrations, denoted by {P_n}_{n=0}^\\infty and {N_n}_{n=0}^\\infty respectively. In particular, we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball with only positive (resp. negative) kinks in the base-level kinky disk, then K is in P_n (resp. N_n). En route to this result we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and is therefore, n-solvable (this also implies that topologically slice knots bound arbitrarily tall gropes in the 4-ball). We also define a variant of Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it is n-solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K is in P_n (resp. N_n). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot which is not topologically slice but lies in each F_n. We also give a 3-dimensional characterization, up to concordance, of knots which bound kinky disks in the 4-ball with only positive (resp. negative) kinks; such knots form a subset of P_0 (resp. N_0).",
      "comment": "30 pages, 21 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-28T21:48:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "smooth knot concordance group",
      "casson tower",
      "filtration",
      "knots bound arbitrarily tall",
      "bound arbitrarily tall gropes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.7532R"
    }
  }
}