Curves on the torus with few intersections
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 . 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.
Keywords
Cite
@article{arxiv.2412.18002,
title = {Curves on the torus with few intersections},
author = {Igor Balla and Marek Filakovský and Bartłomiej Kielak and Daniel Kráľ and Niklas Schlomberg},
journal= {arXiv preprint arXiv:2412.18002},
year = {2026}
}
Comments
Version 2 was a major update of the original version - we determined the maximum exactly for all k (instead of for sufficiently large k as in the original version). Version 4 added a discussion on the column number problem