{
  "id": "2412.18002",
  "version": "v1",
  "published": "2024-12-23T21:51:56.000Z",
  "updated": "2024-12-23T21:51:56.000Z",
  "title": "Curves on the torus with few intersections",
  "authors": [
    "Igor Balla",
    "Marek Filakovský",
    "Bartłomiej Kielak",
    "Daniel Kráľ",
    "Niklas Schlomberg"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "abstract": "Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569--584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most $k$ times, has a maximum size of $k+O(\\sqrt{k}\\log k)$. We prove the maximum size of such a set is at most $k+O(1)$ and determine the exact maximum size for all sufficiently large $k$. In particular, we show that the maximum does not exceed $k+4$ when $k$ is large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-23T21:51:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K20",
      "52A38",
      "90C05"
    ],
    "keywords": [
      "intersections",
      "cambridge philos",
      "simple closed curves",
      "non-homotopic",
      "exact maximum size"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}