Curves on the torus intersecting at most k times

Abstract

AbstractWe 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.