We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves which can be placed on a closed surface of genus \(g\) such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.
KeywordsSimple curves Surfaces Designs
Mathematics Subject Classification57M99 05B99
JM, IR, and LT received support for this work from Rivin’s NSF CDI-I grant DMR 0835586. LT’s final preparation was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 247029-SDModels. Our initial work on this problem took place at Ileana Streinu’s 2010 Barbados workshop at the Bellairs Research Institute.
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