{
  "id": "0912.5348",
  "version": "v1",
  "published": "2009-12-29T17:58:55.000Z",
  "updated": "2009-12-29T17:58:55.000Z",
  "title": "Free Knots and Parity",
  "authors": [
    "Vassily Olegovich Manturov"
  ],
  "comment": "18 pages;12 Figures",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "We consider knot theories possessing a {\\em parity}: each crossing is decreed {\\em odd} or {\\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-triviality theorems for knots from these classes, and constructing maps from knots to knots. Our main example is virtual knot theory and its simplifaction, {\\em free knot theory}. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture), and construct simple and deep invariants made out of parity. Some invariants are valued in graph-like objects and some other are valued in groups. We discuss applications of parity to virtual knots and ways of extending well-known invariants. The existence of a non-trivial parity for classical knots remains an open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-29T17:58:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "invariants",
      "non-trivial free knots",
      "virtual knot theory",
      "free knot theory",
      "open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.5348O"
    }
  }
}