{
  "id": "2204.01789",
  "version": "v1",
  "published": "2022-04-04T18:36:15.000Z",
  "updated": "2022-04-04T18:36:15.000Z",
  "title": "The Number of Closed Essential Surfaces in Montesinos Knots with Four Rational Tangles",
  "authors": [
    "Brannon Basilio"
  ],
  "comment": "39 pages and 34 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we investigate the number of closed, connected, essential, orientable surfaces of a fixed genus $g$, up to isotopy. We show that there are exactly 12 genus 2 surfaces and $8\\phi(g - 1)$ surfaces of genus greater than 2, where $\\phi(g - 1)$ is the Euler totient function of $g - 1$. Observe that this count is independent of the number of crossings of the knot. Moreover, this class of knots form an infinite class of hyperbolic 3-manifolds and the result applies to all such knot complements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T18:36:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K32"
    ],
    "keywords": [
      "closed essential surfaces",
      "rational tangles",
      "hyperbolic montesinos knot",
      "euler totient function",
      "result applies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}