Curves on the torus with few intersections
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914240531529728 |
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| author | Balla, Igor Filakovský, Marek Kielak, Bartłomiej Kráľ, Daniel Schlomberg, Niklas |
| author_facet | Balla, Igor Filakovský, Marek Kielak, Bartłomiej Kráľ, Daniel Schlomberg, Niklas |
| contents | Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$. We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large.
As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18002 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curves on the torus with few intersections Balla, Igor Filakovský, Marek Kielak, Bartłomiej Kráľ, Daniel Schlomberg, Niklas Combinatorics Geometric Topology 57K20, 52A38, 90C05 Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$. We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices. |
| title | Curves on the torus with few intersections |
| topic | Combinatorics Geometric Topology 57K20, 52A38, 90C05 |
| url | https://arxiv.org/abs/2412.18002 |