{
  "id": "1210.0453",
  "version": "v1",
  "published": "2012-10-01T16:11:15.000Z",
  "updated": "2012-10-01T16:11:15.000Z",
  "title": "Complementary Regions of Multi-Crossing Projections of Knots",
  "authors": [
    "MurphyKate Montee"
  ],
  "comment": "15 pages, 21 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "An increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that (2,4,5) and (3,4,n) (where n is a positive integer greater than 4), among others, are universal. In a forthcoming paper, Adams introduces the notion of a multi-crossing projection of a knot. An n-crossing projection} is a projection of a knot in which each crossing has n strands, rather than 2 strands as in a regular projection. We then extend the notion of universality to such knots. These results allow us to prove that (1,2,3,4) is a universal sequence for both n-crossing knot projections, for all n>2. Adams further proves that all knots have an n-crossing projection for all positive n. Another proof of this fact is included in this paper. This is achieved by constructing n-crossing template knots, which enable us to construct multi-crossing projections with crossings of any multiplicity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-01T16:11:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multi-crossing projection",
      "complementary regions",
      "n-crossing projection",
      "integer greater",
      "constructing n-crossing template knots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.0453M"
    }
  }
}