{
  "id": "2003.01007",
  "version": "v1",
  "published": "2020-03-02T16:35:43.000Z",
  "updated": "2020-03-02T16:35:43.000Z",
  "title": "Relating a Bott-Cattaneo-Rossi invariant to Alexander polynomials",
  "authors": [
    "David Leturcq"
  ],
  "comment": "48 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "In a previous article, we gave a more flexible definition of an invariant $(Z_k)_{k\\in \\mathbb N\\setminus\\{0\\}}$ of Bott, Cattaneo, and Rossi defined using integration on configuration spaces for long knots $\\mathbb R^n\\hookrightarrow\\mathbb R^{n+2}$, for odd $n\\geq 3$. This extended the definition of the invariant $(Z_k)_{k\\in\\mathbb N\\setminus\\{0\\}}$ to all long knots in asymptotic homology $\\mathbb R^{n+2}$, for odd $n\\geq3$. In this article, we obtain a formula for $Z_2$ in terms of linking numbers of some cycles of a surface bounded by the knot, when $n\\equiv \\mod 4$. This yields a relation between $Z_2$ and Alexander polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-02T16:35:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q45",
      "57M27",
      "55R80"
    ],
    "keywords": [
      "alexander polynomials",
      "bott-cattaneo-rossi invariant",
      "long knots",
      "configuration spaces",
      "asymptotic homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}