Crossing number inequalities for curves on surfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866913770467491840 |
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| author | Hubard, Alfredo Parlier, Hugo |
| author_facet | Hubard, Alfredo Parlier, Hugo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crossing number inequalities for curves on surfaces Hubard, Alfredo Parlier, Hugo Geometric Topology Computational Geometry Combinatorics 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. |
| title | Crossing number inequalities for curves on surfaces |
| topic | Geometric Topology Computational Geometry Combinatorics |
| url | https://arxiv.org/abs/2504.00916 |