Crossing numbers of simplicial complexes

Alfredo Hubard; Cheikh Saliou Ndiaye; Oriol Solé-Pi

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arXiv · 2610.21017

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 $\mathbb R^{2d}$ 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.

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BibTeX RIS

Alfredo Hubard, Cheikh Saliou Ndiaye, Oriol Solé-Pi. 2026-10-14. Crossing numbers of simplicial complexes. https://arxiv.org/abs/2610.21017

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