{
  "id": "0808.1432",
  "version": "v2",
  "published": "2008-08-11T00:18:12.000Z",
  "updated": "2010-01-17T17:27:27.000Z",
  "title": "Derivatives of Knots and Second-order Signatures",
  "authors": [
    "Tim Cochran",
    "Shelly Harvey",
    "Constance Leidy"
  ],
  "comment": "40 pages, 22 figures, typographical corrections, to appear in Alg. Geom. Topology",
  "journal": "Alg. Geom. Topology 10 (2010), 739-787",
  "doi": "10.2140/agt.2010.10.739",
  "categories": [
    "math.GT"
  ],
  "abstract": "We define a set of \"second-order\" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be \"first-order signatures\". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface, there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new equivalence relation, generalizing homology cobordism, called null-bordism.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-01-17T17:27:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M10"
    ],
    "keywords": [
      "second-order signatures",
      "slice knot",
      "first-order signature",
      "vanishing zero-th order signature",
      "homologically essential simple closed curve"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.1432C"
    }
  }
}