Universal families of arcs and curves on surfaces

TL;DR

This paper studies the minimal size of universal families of curves on surfaces that can realize all pants decompositions through homeomorphisms, providing bounds for surfaces with and without punctures, and extending to polygon triangulations.

Contribution

It introduces the concept of universal families of curves on surfaces, establishing bounds on their minimal size for various surface types and extending the idea to polygon triangulations.

Findings

Exponential upper bound for surfaces without punctures

Superlinear lower bound for surfaces without punctures

Bounds for surfaces with punctures and polygon triangulations

Abstract

The main goal of this paper is to investigate the minimal size of families of curves on surfaces with the following property: a family of simple closed curves $\Gamma$ on a surface realizes all types of pants decompositions if for any pants decomposition of the surface, there exists a homeomorphism sending it to a subset of the curves in $\Gamma$. The study of such universal families of curves is motivated by questions on graph embeddings, joint crossing numbers and finding an elusive center of moduli space. In the case of surfaces without punctures, we provide an exponential upper bound and a superlinear lower bound on the minimal size of a family of curves that realizes all types of pants decompositions. We also provide upper and lower bounds in the case of surfaces with punctures which we can consider labelled or unlabelled, and investigate a similar concept of universality for triangulations of polygons, where we provide bounds which are tight up to logarithmic factors.