Curves on the torus with few intersections
Igor Balla, Marek Filakovsk\'y, Bart{\l}omiej Kielak, Daniel Kr\'a\v{l}, Niklas Schlomberg

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 . 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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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation · Algebraic Geometry and Number Theory
