{
  "id": "2505.09736",
  "version": "v1",
  "published": "2025-05-14T18:54:34.000Z",
  "updated": "2025-05-14T18:54:34.000Z",
  "title": "Taut fillings of the 2-sphere",
  "authors": [
    "Peter Doyle",
    "Matthew Ellison",
    "Zili Wang"
  ],
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "Let $\\sigma$ be a simplicial triangulation of the 2-sphere, $X$ the associated integral 2-cycle. A filling of $X$ is an integral 3-chain $Y$ with $\\partial Y = X$; a taut filling is one with minimal $L_1$-norm. We show that any taut filling arises from an extension of $\\sigma$ to a shellable simplicial triangulation of the 3-ball. The key to the proof is the general fact that any taut filling of an $n$-cycle splits under disjoint union, connected sum, and more generally what we call almost disjoint union, where summands are supported on sets that overlap in at most $n+1$ vertices. Despite the generality of this result, we have nothing to say about optimal fillings of spheres of dimension 3 or higher.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-14T18:54:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "disjoint union",
      "taut filling arises",
      "general fact",
      "cycle splits",
      "optimal fillings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}