On curves intersecting at most once, II

2018/11/04 by Joshua Evan Greene, Greene, Joshua Evan
Computer Science · Mathematics · #05C62 #05D40 #57M15 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Topological and Geometric Data Analysis #math.CO #math.GT #msc:05C62 #msc:05D40 #msc:57M15

paper · pdf · doi:10.48550/arxiv.1811.01413

10 pages

arxiv created 2018/11/04 · openalex publication_date 2018/11/04 · arxiv updated 2018/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that on a closed, orientable surface of genus g, a set of simple loops with the property that no two are homotopic or intersect in more than k points has cardinality \lesssim_k g^k+1 log g. The bound matches the size of the largest known construction to within a factor of ∼_k log g. It generalizes an earlier result of the author, which treated the case k=1. The proof blends probabilistic ideas with covering space arguments related to the fact that surface groups are LERF.

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