{
  "id": "1405.1956",
  "version": "v3",
  "published": "2014-05-08T15:03:30.000Z",
  "updated": "2015-04-13T00:03:58.000Z",
  "title": "Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial",
  "authors": [
    "Dror Bar-Natan",
    "Zsuzsanna Dancso"
  ],
  "comment": "55 pages. This paper is part 1 of a 4-part series whose first two parts originally appeared as a combined preprint, arXiv:1309.7155. April 2015: many minor changes following a referee report",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "This is the first in a series of papers studying w-knotted objects (w-knots, w-braids, w-tangles, etc.), which make a class of knotted objects which is {w}ider but {w}eaker than their usual counterparts. The group of w-braids was studied (as \"{w}elded braids\") by Fenn-Rimanyi-Rourke and was shown to be isomorphic to the McCool group of \"basis-conjugating\" automorphisms of a free group Fn. Brendle-Hatcher, tracing back to Goldsmith, have shown this group to be a group of movies of flying rings in R3. Satoh studied several classes of w-knotted objects (as \"{w}eakly-virtual\") and has shown them to be closely related to certain classes of knotted surfaces in R4. So w-knotted objects are algebraically and topologically interesting. Here we study finite type invariants of w-knotted objects. Following Berceanu-Papadima, we construct homomorphic universal finite type invariants (\"expansions\") of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is essentially the Alexander polynomial. We find that the spaces Aw of \"arrow diagrams\" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic expansion of w-knotted foams is essentially the same as a solution of the Kashiwara-Vergne conjecture (KV), thus giving a topological explanation to the work of Alekseev-Torossian work on KV and Drinfel'd associators. The true value of w-knots, though, is likely to emerge later, for we expect them to serve as a {w}armup example for the study of virtual knots. We expect v-knotted objects to provide the global context whose associated graded structure will be the Etingof-Kazhdan theory of quantization of Lie bialgebras.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-09T01:27:33.000Z",
      "comment": "55 pages. This paper is part 1 of a 4-part series whose first two parts originally appeared as a combined preprint, arXiv:1309.7155",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-13T00:03:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "alexander polynomial",
      "concerning w-knotted objects turn",
      "construct homomorphic universal finite type",
      "homomorphic universal finite type invariants",
      "lie algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1956B"
    }
  }
}