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.
Categories: math.GT math.CO