2-systems of arcs on spheres with prescribed endpoints

Abstract

Let $S$ be an $n$-punctured sphere, with $n\ge 3$. We prove that $\left(\genfrac{}{}{0ex}{}{n}{3}\right)$ 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.

21 pages, 22 figures