{
  "abstract": "Let $E\\xrightarrowπB$ be an oriented circle bundle over an oriented closed surface, and let $\\mathcal E$ denote its Euler number. A \\textit{quasisection} is a smooth surface $Q$, possibly with boundary, together with a generic smooth map $q:Q\\to E$ such that $π\\circ q(Q)=B$. The \\textit{local formula} expresses the Euler number $\\mathcal E$ of a bundle as a sum of the weights of the \\textit{singular vertices} of a quasisection. In the present paper determine the minimal number $\\mathcal N$ of singular vertices for closed and for bordered quasisections for a given circle bundle. For bordered quasisections, \\[ \\mathcal N_{\\mathrm b}(E\\to B) =\\left\\lceil\\frac{3|\\mathcal E|}{2}\\right\\rceil. \\] For closed quasisections, If $\\mathcal{E}$ is even, \\[ \\mathcal N_{\\mathrm cl}(E\\to B) =\\frac{3|\\mathcal E|}{2}. \\] If $\\mathcal{E}$ is odd, \\[ \\mathcal N_{\\mathrm cl}(E\\to B) =4|\\mathcal E|. \\] We also provide examples of quasisections that realize these minimal values.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00679",
  "author_total": 3,
  "authors": [
    {
      "name": "Ivan Kiselev"
    },
    {
      "name": "Gaiane Panina"
    },
    {
      "name": "Timur Shamazov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rjjrrwz157es.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00679",
  "fulltext_url": null,
  "id": "0rjjrrwz157es",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00679",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00679",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rjjrrwz157es.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00679",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00679"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Minimal quasisections of circle bundles",
  "title_text": "Minimal quasisections of circle bundles",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/0rjjrrwz157es",
  "venue": "arXiv",
  "versions": []
}