{
  "id": "1709.10332",
  "version": "v1",
  "published": "2017-09-28T17:41:53.000Z",
  "updated": "2017-09-28T17:41:53.000Z",
  "title": "Knot polynomials from 1-cocycles",
  "authors": [
    "Thomas Fiedler"
  ],
  "comment": "103 pages, 71 figures. arXiv admin note: text overlap with arXiv:1405.5562",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in $H^1(M_n; \\mathbb{Z} [x_1,x_2,...,x_1^{-1},x_2^{-1},...])$, where the number of variables $x_i$ depends on $n$. To each generic point in $M_n$ we associate in a canonical way an arc {\\em scan} in $M_n$, such that $R_n(scan)$ is already a polynomial knot invariant. We show that $R_3(scan)$ detects the non-invertibility of the knot $8_{17}$ in a very simple way and without using the knot group. There are two well-known canonical loops in $M_n$ for each parallel n-cable of a long framed knot $K$: Gramain's loop {\\em rot} and the Fox-Hatcher loop {\\em fh}. The calculation of $R_n$ is of at most quartic complexity for these loops with respect to the number of crossings of $K$ for each fixed $n$. It follows from results of Hatcher that $K$ is not a torus knot if the rational function $R_n(fh(K))/R_n(rot(K))$ is not constant for each $n>1$. $ \\oplus_n R_n$ is a natural candidate in order to separate all classes in $H_1(M_1;\\mathbb{Q}) \\cong H_1(M_n;\\mathbb{Q})$, and in particular to distinguish all knot types $\\pi_0(M_1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-28T17:41:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "knot polynomials",
      "non trivial class",
      "polynomial knot invariant",
      "natural number",
      "generic point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 103,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}