2-Systems of Arcs on Spheres with Prescribed Endpoints

Abstract

Let S be an n-punctured sphere with n≥3. We prove that n3 is the maximum size of a family of pairwise nonhomotopic 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.