Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 4 / c

Choose an essential returning arc $\alpha$ based at $p$ and an essential simple closed curve $\gamma$ with $i(\alpha ,\gamma )>0$. The iteratesare again simple proper arcs based at $p$. Their geometric intersection number with a fixed transverse arc grows linearly with $|n|$, so they represent infinitely many isotopy classes. This is the standard Dehn twist construction.

There are exactly two $\mathrm{Mod}(S_{0,4})$-orbits of vertices. Equality or inequality of the two endpoints is preserved by every homeomorphism. Conversely, a homeomorphism can send any ordered configuration of punctures and complementary discs of an arc to any other of the same endpoint type. Thus all arcs joining distinct punctures lie in one orbit, and all arcs returning to one puncture lie in the other. These are the arc-complex vertex orbits of the four-punctured sphere.