Packing curves on surfaces with few intersections

TL;DR

This paper improves bounds on the maximum number of simple closed curves with limited intersections on surfaces, showing polynomial growth with sharper estimates, especially for the case of one intersection.

Contribution

It narrows Przytycki's bounds by providing tighter asymptotic estimates for the size of maximal curve collections with bounded intersections on surfaces.

Findings

Maximal collections grow as O(|χ|^{3k}/(log|χ|)^2)

For fixed genus, the growth is at most O(n^{2k+2})

Exact growth rate for k=2 is Θ(n^3)

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 $$ \mathcal{N}_{k}(S) \leq O(n^{2k+2}) . $$ Using similar techniques, we also obtain the sharp estimate $\mathcal{N}_{2}(S)=\Theta(n^3)$ when $k=2$ and $g$ is fixed.