# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.06145