Curves on the torus intersecting at most k times

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 \(...

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Bibliographic Details
Published in:arXiv.org 2020-08
Main Authors: Aougab, Tarik, Gaster, Jonah
Format: Article
Language:English
Subjects:
Combinatorial analysis
Toruses
Online Access:Get full text
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Summary: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.
ISSN:2331-8422