I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism (i.e if it is periodic)?
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If your question is: can you present the homeotopy group in terms of generators and relations, the answer is "yes", following the work of HatcherThurston, Wajnryb, and most recently M. Korkmaz, who gives a relatively civilized presentation. If you mean: given a homeomorphism, can you express its isotopy class in terms of the generators, I assume that the answer is yes, but it obviously depends on how the homeomorphism is given. For related work, see Brinkmann, Peter(1UT) An implementation of the BestvinaHandel algorithm for surface homeomorphisms. (English summary) Experiment. Math. 9 (2000), no. 2, 235–240. 

