2-systems of arcs on spheres with prescribed endpoints

TL;DR

This paper establishes the maximum size of a family of simple arcs on an n-punctured sphere with prescribed endpoints, where arcs are pairwise non-homotopic and intersect at most twice, and explores related diagram properties.

Contribution

It proves the maximum size of such arc families and analyzes properties of square annular diagrams with simple arcs and limited intersections.

Findings

Maximum size of arc families is inom{n}{3} for n t least 3.

Square annular diagrams with at least one square have a corner on each boundary path.

Dual curves of these diagrams are simple arcs intersecting at most once.

Abstract

Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on each of its boundary paths if $A$ contains at least one square and the dual curves of $A$ are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most once.