Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary

TL;DR

This paper establishes an upper bound on the size of families of essential simple arcs on an n-punctured disk that intersect at most twice, and explores related properties of certain square complexes.

Contribution

It introduces a new combinatorial bound for arcs on punctured disks and analyzes the structure of related square complexes.

Findings

Maximum size of arc families is inom{n+1}{3}

Arc intersections are limited to at most twice

Square complexes with such arcs must have a corner or spur

Abstract

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.