Crossing number inequalities for curves on surfaces

Alfredo Hubard; Hugo Parlier

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arXiv · 2504.00916

Crossing number inequalities for curves on surfaces

Alfredo Hubard·Hugo Parlier

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Abstract

We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m \log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

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Alfredo Hubard, Hugo Parlier. 2025-04-01. Crossing number inequalities for curves on surfaces. https://arxiv.org/abs/2504.00916

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