From arcs to curves: quadratic growth of 1-systems

TL;DR

This paper proves that the maximum size of a collection of simple closed curves intersecting at most once on an orientable surface grows quadratically with the surface's Euler characteristic, resolving a longstanding open problem.

Contribution

It introduces new concepts like almost nibs, flowers, and stem systems to analyze curve distributions, providing a quadratic growth bound.

Findings

Largest 1-system size grows quadratically with surface complexity

Resolved a longstanding question by Farb-Leininger

Introduced novel geometric concepts for curve analysis

Abstract

We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $\chi$ grows quadratically in $|\chi|$. This resolves a longstanding question of Farb-Leininger, up to multiplicative constants. Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of \textit{almost nibs}, \textit{flowers}, and \textit{stem systems} in order to account for how certain polygons built from pairs of curves in the collection distribute their area over the surface.