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.

Arc-complex vertex orbits of the four-punctured sphere

Choose an essential returning arc $\alpha$ based at $p$ and an essential simple closed curve $\gamma$ with $i(\alpha,\gamma)>0$. The iterates
$$
T_\gamma^n(\alpha),\qquad n\in\mathbb Z,
$$
are 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 $\operatorname{Mod}(S_{0,4})$-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>.

Solution

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

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A Dehn twist $T_\gamma$ cuts an oriented surface along a simple closed curve $\gamma$, rotates one side once, and glues it back. If a curve or arc $\alpha$ has nonzero geometric intersection with $\gamma$, the iterates $T_\gamma^n(\alpha)$ usually represent distinct isotopy classes, with intersection numbers growing linearly in $|n|$.


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