{
  "id": "1908.01942",
  "version": "v1",
  "published": "2019-08-06T03:45:06.000Z",
  "updated": "2019-08-06T03:45:06.000Z",
  "title": "A lower bound on critical points of the electric potential of a knot",
  "authors": [
    "Max Lipton"
  ],
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "Given a knot $K$ parametrized by $r: [0,2\\pi] \\to \\mathbb{R}^3$, we can define the electric potential on its complement by $\\Phi(x) = \\int_0^{2\\pi} \\frac{|r'(t)|}{|x - r(t)|}dt$. Physicists and knot theorists want to understand the critical points of the potential and their behavior. The tunneling number $t(K)$ of a knot is the smallest number of arcs one needs to add to a knot so the complement is a handlebody. We show the number of critical points of the potential is at least $2t(K) + 2$. The result is proven using Morse theory and stable manifold theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-06T03:45:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical points",
      "electric potential",
      "lower bound",
      "knot theorists want",
      "morse theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}