Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The group of diffeomorphisms preserving the marked points acts on this set. What are the orbits? How many are there? Equivalently, we can instead consider embedded arcs up to isotopy and consider the action of the mapping class group of the surface. 


I am specifically interested in two cases: genus zero with four marked points, and genus 1 with two marked points. However I think the general question is also interesting and it seems like the sort of thing that has been studied before. I just no idea where to look. 



Is the Action of the mapping class group transitive on embedded arcs?
There is only one orbit. 


Suppose you have two such arcs $\lambda, \lambda'$. Let $S\_\lambda$ be obtained from $S$ by removing a regular neighborhood of $\lambda$: the regular neighborhood is a disc, containing the two marked points in its interior. The two surfaces $S\_\lambda$ and $S\_{\lambda'}$ so obtained have the same Euler characteristic and the same number (one) of boundary components, hence they are diffeomorphic. 


You can arrange the diffeomorphism so that it preverves the other marked points (simply move them by isotopy). You can then extend it to the removed discs and get a diffeomorphism of $S$ preserving the punctures and sending $\lambda$ to $\lambda'$.


