I need to know if there is an algortihm to write down a Dehn twist about an arbitrary curve on an orientable surface $S$ , as product of a set of generators of $MCG(S)$.

Since we have the conjugacy relation: $\phi^{-1}D_a\phi=D_{\phi(a)}$ ($D$ being the Dehn twist and equality understood up to isotopy), it will also help to know if for an arbitrary curve $C$ we can find a generator $a$(from any set of generators) and a diffeomorphism $\phi$ such that $\phi(a)=C$.

do, though. $\endgroup$