{
  "id": "2009.04637",
  "version": "v2",
  "published": "2020-09-10T02:18:34.000Z",
  "updated": "2022-01-29T01:06:10.000Z",
  "title": "Totally geodesic surfaces in twist knot complements",
  "authors": [
    "Khanh Le",
    "Rebekah Palmer"
  ],
  "comment": "v2. Corrected typos, included Magma and SageMath codes in ancillary files. 25 pages, 6 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher, Miller and Stover. The construction comes from a family of twist knot complements and their dihedral covers. The case $k=1$ arises from the uniqueness of an immersed totally geodesic thrice-punctured sphere, answering a question of Reid. Applying the proof techniques of the main result, we explicitly construct non-elementary maximal Fuchsian subgroups of infinite covolume within twist knot groups, and we also show that no twist knot complement with odd prime half twists is right-angled in the sense of Champanerkar, Kofman, and Purcell.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-29T01:06:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "twist knot complement",
      "totally geodesic surfaces",
      "totally geodesic thrice-punctured sphere",
      "construct non-elementary maximal fuchsian subgroups",
      "explicitly construct non-elementary maximal fuchsian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}