Curves on the torus with prescribed intersections

Juvan, Malnič and Mohar [J. Combin. Theory Ser. B 68 (1996), 7-22] proved that for every surface S and every integer k, there exists N(S,k) such that any set of simple closed curves on the surface S, where any two are non-homotopic and intersect at most k times, has a maximum size at most N(S,k). In the case when S is the torus T^2, the problem has an interesting connection to number theory as the Riemann hypothesis yields that N(T^2,k)<=k+O(k^0.5*log k); Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] have recently proven that this bound holds directly. We determine N(T^2,k) exactly for every k. In particular, we show that N(T^2,k)<=k+6 for all k and N(T^2,k)<=k+4 when k is sufficiently large.

The talk is based on joint work with Igor Balla, Marek Filakovský, Bartłomiej Kielak and Niklas Schlomberg.