Labeled double pants decompositions

A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in [3]. There are two natural operations acting on double pants decompositions: flips and handle-twists. It is shown in [3] that the groupoid generated by flips and handle-twists acts transitively on admissible double pants decompositions where the class of admissible decompositions has a natural topological and combinatorial description. In this paper, we label the curves of double pants decompositions and show that for all but one surfaces the same groupoid acts transitively on all labeled admissible double pants decompositions. The only exclusion is a sphere with two handles, where the groupoid has 15 orbits.