Double pants decompositions revisited

Double pants decompositions were introduced in our paper "Double pants decompositions of 2-surfaces" (Mosc. Math. J. 11 (2011), no. 2, 231-258, arXiv:1005.0073), together with a flip-twist groupoid acting on these decompositions. It was shown that flip-twist groupoid acts transitively on a certain topological class of the decompositions, however, recently Randich discovered a serious mistake in the proof. In this note we present a new proof of the result, accessible without reading the initial paper.