{
  "id": "2406.17738",
  "version": "v1",
  "published": "2024-06-25T17:24:24.000Z",
  "updated": "2024-06-25T17:24:24.000Z",
  "title": "Average signature and 4-genus of 2-bridge knots",
  "authors": [
    "Moshe Cohen",
    "Adam M. Lowrance",
    "Neal Madras",
    "Steven Raanes"
  ],
  "comment": "33 pages, 10 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\\sqrt{{2c}/{\\pi}}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\\log c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-25T17:24:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "05A05",
      "60J10"
    ],
    "keywords": [
      "average signature",
      "specific sublinear upper bound",
      "diagrammatic crossing number",
      "expected absolute value",
      "knot diagrams"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}