{
  "id": "1610.06514",
  "version": "v1",
  "published": "2016-10-20T17:39:22.000Z",
  "updated": "2016-10-20T17:39:22.000Z",
  "title": "Packing curves on surfaces with few intersections",
  "authors": [
    "Tarik Aougab",
    "Ian Biringer",
    "Jonah Gaster"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "Przytycki has shown that the size $\\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|\\chi(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$ \\mathcal{N}_{k}(S) =O \\left( \\frac{ |\\chi|^{3k}}{ ( \\log |\\chi| )^2 } \\right) , $$ In particular, the size of a maximal 1-system grows sub-cubically in $|\\chi(S)|$. The proof uses a circle packing argument of Aougab-Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we can improve our estimates using a different argument to give $$ \\mathcal{N}_{k}(S) \\leq O(n^{2k+2}) . $$ Using similar techniques, we also obtain the sharp estimate $\\mathcal{N}_{2}(S)=\\Theta(n^3)$ when $k=2$ and $g$ is fixed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-20T17:39:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "packing curves",
      "intersections",
      "narrow przytyckis bounds",
      "sharp estimate",
      "similar techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}