{
  "id": "1511.05624",
  "version": "v1",
  "published": "2015-11-17T23:59:11.000Z",
  "updated": "2015-11-17T23:59:11.000Z",
  "title": "Finite Type Invariants of w-Knotted Objects IV: Some Computations",
  "authors": [
    "Dror Bar-Natan"
  ],
  "comment": "About 49 pages. The version at http://www.math.toronto.edu/~drorbn/LOP.html#WKO4 may be more recent",
  "categories": [
    "math.GT"
  ],
  "abstract": "In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), Z. Dancso and I studied a certain theory of \"homomorphic expansions\" of \"w-knotted objects\", a certain class of knotted objects in 4-dimensional space. When all layers of interpretation are stripped off, what remains is a study of a certain number of equations written in a family of spaces $A^w$, closely related to degree-completed free Lie algebras and to degree-completed spaces of cyclic words. The purpose of this paper is to introduce mathematical and computational tools that enable explicit computations (up to a certain degree) in these $A^w$ spaces and to use these tools to solve the said equations and verify some properties of their solutions, and as a consequence, to carry out the computation (up to a certain degree) of certain knot-theoretic invariants discussed in [WKO1]-[WKO3] and in my related paper [KBH] (arXiv:1308.1721).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-17T23:59:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "finite type invariants",
      "w-knotted objects",
      "degree-completed free lie algebras",
      "cyclic words",
      "knot-theoretic invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151105624B"
    }
  }
}