{
  "id": "1301.6298",
  "version": "v2",
  "published": "2013-01-26T23:40:26.000Z",
  "updated": "2014-07-23T21:48:44.000Z",
  "title": "Denominators and Differences of Boundary Slopes for (1,1)-Knots",
  "authors": [
    "Jason Callahan"
  ],
  "comment": "11 pages, 5 figures. Results added and generalized, paper restructured and exposition improved",
  "categories": [
    "math.GT"
  ],
  "abstract": "We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers m, n > 1 with n odd the exterior of the Montesinos knot K(-1/2, m/(2m \\pm 1), 1/n) in S^3 contains an essential surface with boundary slope r = 2(n-1)^2/n if m is even and 2(n+1)^2/n if m is odd. If n > 4m, we prove that K(-1/2, m/(2m+1), 1/n) also has a boundary slope whose difference with r is (8m-2)/(n^2-4mn+n), which decreases to 0 as n increases. All of these knots are (1,1)-knots.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-23T21:48:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25"
    ],
    "keywords": [
      "boundary slope",
      "denominator",
      "nonzero integer occurs",
      "arbitrarily small difference",
      "montesinos knot"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.6298C"
    }
  }
}