I am a physicist who's getting started with Mapping Class Group for Riemann surfaces, pants decompositions and triangulations so I apologise in advance if the following is a stupid question/wrong.
I understand that to any pants decomposition of a Riemann surface we can associate a set of generators (the Dehn twists). Different pants decompositions gives different sets of generators, and relations among various sets of generators are understood as being generated by a minimal sets of relations (Lantern, Chain, Braiding...)
My question is: is there a similar picture for Triangulations? Given a Triangulation, can I assign a canonical set of generators of the Mapping Class Group? Can I understand relations between generators using flips of triangulations?