From arcs to curves: quadratic growth of 1-systems

Abstract

We show that a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $\chi$ has at most $2016|\chi {|}^2+338|\chi |$ curves. Up to multiplicative constants, this resolves a thirty-year old problem (see Problem 2.12(b) from the K3 Problem List). Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of tulips, flowers, and stem systems in order to account for how certain polygons built from pairs of curves in the collection distribute area over the surface.

24 pages, 19 figures. Version 2 has a subsantially rewritten sections 4 and 5, as well as redone Proposition 2.14. As a result, constants in the main theorem have gotten worse. (In addition to begin correct, the new version will apply more directly to $k>1$.)