Curves on the torus intersecting at most k times

TL;DR

This paper establishes a new upper bound on the number of simple closed curves on a torus that intersect at most k times, improving previous bounds through elementary combinatorial and geometric methods.

Contribution

It provides an improved, elementary bound of O(√k log k) for the maximum size of such curve sets, surpassing prior complex bounds involving prime gaps and the Riemann hypothesis.

Findings

Bound of k + O(√k log k) on the size of curve sets

Elementary combinatorial and geometric proof techniques

Improved understanding of curve intersection limits on the torus

Abstract

We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k} \log k)$. Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$, and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain $O(\sqrt{k} \log k)$. By contrast, our methods are elementary, combinatorial, and geometric.