{
  "id": "1311.0526",
  "version": "v1",
  "published": "2013-11-03T20:55:56.000Z",
  "updated": "2013-11-03T20:55:56.000Z",
  "title": "Bounds on Übercrossing and Petal Numbers for Knots",
  "authors": [
    "Colin Adams",
    "Orsola Capovilla-Searle",
    "Jesse Freeman",
    "Daniel Irvine",
    "Samantha Petti",
    "Daniel Vitek",
    "Ashley Weber",
    "Sicong Zhang"
  ],
  "comment": "13 pages, 8 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An \\\"ubercrossing projection has a single $n$-crossing and a petal projection has a single $n$-crossing such that there are no loops nested within others. The \\\"ubercrossing number, $\\text{\\\"u}(K)$, is the smallest $n$ for which we can represent a knot $K$ with a single $n$-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the \\\"{u}bercrossing number and petal number to well-known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are tight for $(r, r+1)$-torus knots. We also explore the behavior of \\\"{u}bercrossing number under composition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-03T20:55:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "petal number",
      "minimal petal projection",
      "strand bisects",
      "strands cross",
      "torus knots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0526A"
    }
  }
}