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PreprintCC BY 4.0arXiv

Crossing numbers of simplicial complexes

  • Alfredo Hubard
  • Cheikh Saliou Ndiaye
  • Oriol Solé-Pi
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Abstract

This paper studies crossing numbers of d-dimensional simplicial complexes drawn in ℝ2d or 𝕊2d. 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.

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Cite

BibTeXRIS
BibTeX
@misc{hubard2026crossing,
  title = {{Crossing numbers of simplicial complexes}},
  author = {Alfredo Hubard and Cheikh Saliou Ndiaye and Oriol Solé-Pi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21017},
  eprint = {2610.21017},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21017},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Crossing numbers of simplicial complexes
AU  - Alfredo Hubard
AU  - Cheikh Saliou Ndiaye
AU  - Oriol Solé-Pi
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.21017
UR  - https://arxiv.org/abs/2610.21017
AB  - This paper studies crossing numbers of d-dimensional simplicial complexes drawn in ℝ^2d or 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.
ER  -