The brick polytope of a sorting network

TL;DR

This paper introduces the brick polytope associated with a sorting network, characterizing its structure, faces, and decompositions, and connects it to known polytopes like the associahedron and multitriangulations.

Contribution

It constructs and analyzes the brick polytope of a sorting network, unifying various polytope realizations and exploring their combinatorial and geometric properties.

Findings

Vertices are characterized combinatorially.

The polytope is described as a Minkowski sum of matroid polytopes.

Includes realizations of the associahedron as brick polytopes.

Abstract

The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of flip graphs on pseudoline arrangements with contacts supported by a given sorting network. In this paper, we construct the brick polytope of a sorting network, obtained as the convex hull of the brick vectors associated to each pseudoline arrangement supported by the network. We combinatorially characterize the vertices of this polytope, describe its faces, and decompose it as a Minkowski sum of matroid polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes for certain well-chosen sorting networks. We furthermore discuss the brick polytopes of sorting networks supporting pseudoline arrangements which correspond to multitriangulations of convex polygons: our polytopes only realize subgraphs of the flip graphs on multitriangulations and they cannot appear as projections of a hypothetical multiassociahedron.