Arc-complex vertex orbits of the four-punctured sphere
= Arc-complex vertex orbits of the four-punctured sphere
The mapping class group of the four-punctured sphere has two orbits on vertices of its arc complex: arcs whose endpoints are distinct and arcs whose endpoints coincide. Each orbit is infinite, and <Dehn twist>[Dehn twists] exhibit infinitely many classes of the second type based at any fixed puncture.