$\DeclareMathOperator{\MCG}{\operatorname{MCG}}$Consider a bordered, punctured, orientable surface $S$. Associated to it there is its mapping class group $\MCG(S)$. One way to concretely think about it is in terms of generators and relations, which can be done in many ways. Typically one chooses a set of closed simple curves on $S$, defines associated Dehn twists and half-twists and then gives set of relations among them in terms of how curves intersect on $S$.

Recently I discovered that there is a very different presentation of the mapping class group in terms of its actions on (isotopy classes of) triangulations on the surface, see for example Penner's book on decorated Teichmuller theory. Essentially, the idea is that a MCG element takes a triangulation to another one with the same combinatorics. Since triangulations are acted transitively by a set of moves (flips, quasi-flips and further generalization used in the context of cluster algebras), each mapping class group element is corresponds to a sequence of flips which connects two combinatorially equivalent triangulations. Furthermore, if one adds 2-dimensional faces corresponding to the so-called pentagon identities, the complex whose vertices are (isotopy classes of) triangulations and whose edges are flips one gets a simply connected 2-dim CW-complex.

By choosing any vertex in this complex, one can then think of the mapping class group as the set of paths that join this vertex to any other vertex labelled by a combinatorially equivalent triangulation, modulo the contractible paths (which are a combination of the pentagon identities).

My question is, is there an explicit/algorithmic way to produce a sequence-of-flips representative for a given Dehn twist?

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