{
  "id": "2004.04624",
  "version": "v1",
  "published": "2020-04-09T16:10:48.000Z",
  "updated": "2020-04-09T16:10:48.000Z",
  "title": "More 1-cocycles for classical knots",
  "authors": [
    "Thomas Fiedler"
  ],
  "comment": "85 pages, 65 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $n$ be a natural number and let $M$ be the moduli space of all knots in the solid torus $V^3$ which represent the homology class $n \\in \\mathbb{Z}\\cong H_1(V^3)$. Let $K$ be a framed oriented long knot and let $nK$ be its n-cable twisted by a fixed string link $T$ to a knot in the solid torus $V^3$. Let $M_n^{reg}$ be the topological moduli space of all such knots $nK$ up to regular isotopy with respect to a fixed projection into the annulus. We construct two new sorts of 1-cocycles: two integer 1-cocycles $R_{[a,b,c]}$ and $R_{(a,b,c)}$ for $M$, which use linear weights and which depend on {\\em three} integer parameters $a,b,c \\in \\mathbb{Z}$, and two integer 1-cocycles $R_{a\\pm}^{(2)}$ for $M_n^{reg}$, which use {\\em quadratic} weights and which depend on a natural number $0<a<n$. The Lagrange interpolation polynomials of $R_{a\\pm}^{(2)}(\\gamma)$ are candidates for new polynomial knot invariants for classical knots $K$, which can be calculated with polynomial complexity for each homology class $[\\gamma] \\in H_1(M_n^{reg})$. The 1-cocycles $R_{[a,b,c]}$ and $R_{(a,b,c)}$ are candidates to distinguish the orientations of classical knots.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-09T16:10:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "classical knots",
      "moduli space",
      "solid torus",
      "homology class",
      "natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 85,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}