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arXiv · 2412.18002

Curves on the torus with few intersections

Igor Balla·Marek Filakovský·Bartłomiej Kielak·Daniel Kráľ·Niklas Schlomberg
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Abstract

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.

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Igor Balla, Marek Filakovský, Bartłomiej Kielak, Daniel Kráľ, Niklas Schlomberg. 2024-12-23. Curves on the torus with few intersections. https://arxiv.org/abs/2412.18002

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