{
  "id": "2410.17443",
  "version": "v1",
  "published": "2024-10-22T21:44:57.000Z",
  "updated": "2024-10-22T21:44:57.000Z",
  "title": "Generating Infinitely Many Hyperbolic Knots with Plats",
  "authors": [
    "Carolyn Engelhardt",
    "Seth Hovland"
  ],
  "comment": "15 pages, 12 figures, Comments Welcome!",
  "categories": [
    "math.GT"
  ],
  "abstract": "In this paper we study the relationships between links in plat position, the dynamics of the braid group, and Heegaard splittings of double branched covers of $S^3$ over a link. These relationships offer new ways to view links in plat position and a new tool kit for analyzing links. In particular, we show that the Hempel distance of the Heegaard splitting of the double branched cover obtained from a plat is a lower bound for the Hempel distance of that plat. Using the Hempel distance of a knot in bridge position and pseudo-Anosov braids we obtain our main result: a construction of infinitely many sequences of prime hyperbolic $n$-bridge knots for $n \\geq 3$, infinitely many of which are distinct. We consider known results to show that the knot genus and hyperbolic volume of these knots are bounded below by a linear function. As a further result we show that the plat closure of a 6-braid is generically hyperbolic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-22T21:44:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K20",
      "57K30"
    ],
    "keywords": [
      "hyperbolic knots",
      "hempel distance",
      "plat position",
      "double branched cover",
      "relationships offer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}