# Mapping One Curve to another using Dehn Twists

Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm that will produce a sequence of Dehn twists (preferably about the Lickorish generators) that will map $\alpha$ to $\beta$?

Right now I am trying to map one curve to another on a three hole torus using Dehn twists by hand, but I am just guessing twists and getting nowhere.

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The two curves must be of the same type (have homeomorphic complements) or else you'll be looking forever. –  Richard Kent Oct 29 '13 at 16:58
Lickorish's textbook "An introduction to knot theory" has a chapter on mapping class groups where this sequence of Dehn Twists (for a given set of nonseparating curves) is explicitly produced (and this is afterwards used to prove his theorem about MCG being generated by Dehn Twists) –  ThiKu Oct 29 '13 at 17:14
Look at Lickorish's proof that the mapping class group is finitely generated by Dehn twists. There's a key step where you go from generated by all Dehn twists to a finite collection, that's what you want. It's also in Birman's mapping class groups book. Likely it would also be in Margalit and Farb. –  Ryan Budney Oct 29 '13 at 17:15
@user39082 Thanks! I will take a look at Lickorish's book –  Darren G Oct 29 '13 at 18:50
@RyanBudney Thanks! I am going to take a look at Lickorish's proof –  Darren G Oct 29 '13 at 18:52