Labeled double pants decompositions

TL;DR

This paper studies labeled double pants decompositions on surfaces, showing that a groupoid acts transitively on all but one specific case, revealing detailed symmetry properties of these decompositions.

Contribution

It extends previous work by labeling curves in double pants decompositions and analyzing the resulting groupoid action on these labeled structures.

Findings

Groupoid acts transitively on labeled decompositions for all but one surface

Sphere with two handles has 15 distinct orbits under the groupoid

Provides a detailed topological and combinatorial classification of labeled decompositions

Abstract

A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 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.