{
  "abstract": "This paper studies crossing numbers of $d$-dimensional simplicial complexes drawn in $\\mathbb R^{2d}$ or $\\mathbb S^{2d}$. We clarify and generalize a number of known results and conjectures from the graph case to higher dimensions: the crossing lemma (including the exceptional case of two-dimensional complexes drawn in dimension $4$), a generalized Harary-Hill conjecture (and evidence for it), existence of a midrange crossing constant (in a weak sense), and upper bound constructions based on stochastic geometry (related to Sylvester's problem) and the cyclic polytope. Some of our results are new even in the case of graphs: we give a simple proof of the Harary-Hill conjecture for antipodal drawings, and sharpen the threshold above which the rectilinear and spherical crossing numbers of the Erdős-Rényi random graph behave like those of the complete graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21017",
  "author_total": 3,
  "authors": [
    {
      "name": "Alfredo Hubard"
    },
    {
      "name": "Cheikh Saliou Ndiaye"
    },
    {
      "name": "Oriol Solé-Pi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bkh8pansmns8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21017",
  "fulltext_url": null,
  "id": "0bkh8pansmns8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21017",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7222537810",
  "pdf_url": "https://arxiv.org/pdf/2610.21017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bkh8pansmns8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Crossing numbers of simplicial complexes",
  "title_text": "Crossing numbers of simplicial complexes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/0bkh8pansmns8",
  "venue": "arXiv",
  "versions": []
}