Curves on the torus with few intersections

TL;DR

This paper determines the maximum size of sets of simple closed curves on a torus with limited intersections, providing exact bounds for all intersection counts and linking the problem to a combinatorial matrix column number problem.

Contribution

It precisely characterizes the maximum size of such curve sets for every intersection number, resolving a previously unknown aspect of the problem.

Findings

Maximum size never exceeds k+6

Maximum size does not exceed k+4 for large k

Results settle the column number problem for certain matrices

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.