Packing curves on surfaces with few intersections

Abstract

Przytycki has shown that the size ${\mathcal{N}}_k(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|\chi (S)|$ of degree $k^2+k+1$. In this paper, we narrow Przytycki's bounds by showing that $${\mathcal{N}}_k(S)=O\left(\frac{|\chi {|}^{3k}}{(\log |\chi |{)}^2}\right),$$ In particular, the size of a maximal 1-system grows sub-cubically in $|\chi (S)|$. The proof uses a circle packing argument of Aougab-Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we can improve our estimates using a different argument to give $${\mathcal{N}}_k(S)\le O(n^{2k+2}).$$ Using similar techniques, we also obtain the sharp estimate ${\mathcal{N}}_2(S)=\Theta (n^3)$ when $k=2$ and $g$ is fixed.