Topological Designs

TL;DR

This paper establishes new bounds on the maximum number of simple closed curves with limited intersections on surfaces of various genera, advancing understanding of topological curve arrangements.

Contribution

It provides the first exponential upper and quadratic lower bounds for large genus surfaces, and exact solutions for genus one and two.

Findings

Exponential upper bound on the number of curves for large genus

Quadratic lower bound on the number of curves for large genus

Exact solutions for genus one and two cases

Abstract

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves 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.