{
  "id": "1405.5562",
  "version": "v3",
  "published": "2014-05-21T21:43:19.000Z",
  "updated": "2015-07-05T17:52:51.000Z",
  "title": "Singularization of knots and closed braids",
  "authors": [
    "Thomas Fiedler"
  ],
  "comment": "154 pages, 120 figures. Almost entirely rewritten, results considerably improved, new results added, incorrect examples deleted. arXiv admin note: text overlap with arXiv:1304.0970",
  "categories": [
    "math.GT"
  ],
  "abstract": "We construct the first combinatorial 1-cocycle with values in the $ \\mathbb{Z} [x,x^{-1}]$-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invariant with values in a $ \\mathbb{Z} [x,x^{-1}]$-module generated by 3-manifolds for each element of infinite order of the mapping class group of the complement of a satellite knot in $S^3$. The 1-cocycle seems to be trivial on all loops for long knots which are not satellites but it is already non trivial on the Fox-Hatcher loop of any composite knot, on the loop which consists of {\\em dragging} a trefoil through another trefoil and on the {\\em scan-arc} for the 2-cable of the trefoil. The {\\em canonical resolution} of the value of the 1-cocycle for dragging a knot through another knot leads to a symmetric bilinear form on the free $ \\mathbb{Z} [x,x^{-1}]$-module of all unframed oriented knot types into itself. We conjecture that its radical contains only the trivial knot. Evaluating e.g. the Kauffman-Vogel HOMFLYPT polynomial for singular knots on the value of the 1-cocycle applied to the associated quadratic form, leads to a couple of new 3-variable knot polynomials. We construct also the first non trivial 1-cocycle for those closed positive 4-braids which contain a half-twist. It takes its values in a symmetric power of the $ \\mathbb{Z}$-module of isotopy classes of closed positive 4-braids with a double point.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-10T15:44:04.000Z",
      "abstract": "We give a method to construct non symmetric solutions of a global tetrahedron equation. The solutions gives rise to the first combinatorial 1-cocycles with values in the $ \\mathbb{Z}$-module of singular long knots with signed planar double points and which represent non trivial cohomology classes in the topological moduli space of long knots. The canonical resolution of the evaluation of the 1-cocycles on Hatcher's loop for a knot leads to a {\\em wave} in the space of all long knots. Each knot invariant becomes a new invariant by evaluating it on the wave. In particular, the Kauffman-Vogel HOMFLYPT polynomial for singular knots applied to our 1-cocycles on Hatcher's loop leads to the first quantum knot polynomials which are {\\em not multiplicative} for the connected sum of knots. A variation of the combinatorial method produces universally defined non trivial combinatorial 1-cocycles for all closed n-braids which are knots. We use them to formulate the {\\em entropy conjecture} for pseudo-Anosov braids, which is an analogue of the volume conjecture for hyperbolic knots, and we give some evidence for it.",
      "comment": "165 pages, 152 figures. Missing figures, calculations and references added. arXiv admin note: text overlap with arXiv:1304.0970. text overlap with arXiv:1304.0970",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-07-05T17:52:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "defined non trivial combinatorial",
      "closed braids",
      "produces universally defined non",
      "universally defined non trivial",
      "long knots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 154,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.5562F"
    }
  }
}