Large Collections of Curves Pairwise Intersecting Exactly Once

TL;DR

This paper proves that for surfaces of genus at least 3, there are at least two distinct maximum collections of simple closed curves intersecting exactly once, addressing a question about their classification.

Contribution

It demonstrates the existence of multiple maximum collections of curves with pairwise one intersection on higher genus surfaces, expanding understanding of their structure.

Findings

Existence of at least two such collections for genus ≥ 3

Maximum size of collections is 2g+1

Answer to a question by Malestein, Rivin, and Theran

Abstract

Let $\Omega=(\omega_{j})_{j\in I}$ be a collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus $g$ surface $S_{g}$, such that $\omega_{i}$ and $\omega_{j}$ intersect exactly once for $i\neq j$. It was recently demonstrated by Malestein, Rivin, and Theran that the cardinality of such a collection is no more than $2g+1$. In this paper, we show that for $g\geq 3$, there exists at least two such collections with this maximum size up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran.