Curves on the torus with few intersections

How large can a set of simple closed curves on a torus be, such that any two curves are non-homotopic and intersect at most k times? It is known since 1996 that for any fixed \(k\), such a set must be finite. The topic has been extensively studied, leading to a recent upper bound of \(k + O(k^\frac{1}{2} log k)\) on the size of the set, established by Aougab and Gaster. We resolve the problem by determining the optimal bound and providing a matching construction for every value of \(k\). In particular, we show that the size of such a set never exceeds \(k + 6\), and is at most \(k + 4\) for sufficiently large \(k\). In this talk, we will present the main ideas behind the proof, which utilizes well-known tools from combinatorics, discrete optimization, and geometry, along with some number-theoretic observations.

This is joint work with Igor Balla, Marek Filakovský, Daniel Král’, and Niklas Schlomberg.