Arc complex ($\mathcal{A}(S)$)

The vertices of the arc complex of a punctured surface $S$ are isotopy classes of unoriented essential simple proper arcs. A finite set spans a simplex when its classes have representatives with pairwise disjoint interiors. The mapping class group acts on the complex by simplicial automorphisms.

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.

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 twists exhibit infinitely many classes of the second type based at any fixed puncture.