Laminations of a graph on a pair of pants

TL;DR

This paper characterizes the lamination space of graphs embedded on a pair of pants surface, linking topological graph properties to spectral map images via lattice polytopes.

Contribution

It identifies the lamination space as a lattice polytope and characterizes which polytopes can arise from such graphs on a pair of pants.

Findings

Lamination space forms a lattice polytope.

Characterization of polytopes from graphs on a pair of pants.

Connection to the spectral map of the vector bundle Laplacian.

Abstract

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice polytope, then we characterize the polytopes that arise as the lamination space of some graph on a pair of pants. This characterizes the image of a purely topological version of the spectral map for the vector bundle Laplacian for a flat connection on a pair of pants. The proof uses a graph exploration technique akin to the peeling of planar maps.