Arc complex of the three-punctured sphere ($\mathcal{A}(S_{0,3})$)

The three-punctured sphere has six arc classes: one joining each unordered pair of distinct punctures and one returning to each puncture. The three joining arcs span one triangle; for each puncture, its returning arc and the two joining arcs incident to it span another triangle. Its mapping class group is the symmetric group on the three punctures and has two vertex orbits, distinguished by whether the endpoints agree.