Homotopy type of intervals of the second higher Bruhat orders

TL;DR

This paper investigates the topological structure of intervals within the second higher Bruhat orders, revealing their correspondence with zonogonal tilings and establishing their noncontractibility in certain cases.

Contribution

It proves that noncontractible intervals in the second higher Bruhat order correspond to zonogonal tilings, connecting combinatorial and geometric properties.

Findings

Noncontractible intervals correspond to zonogonal tilings.

The two standard orderings on rhombic tilings are identical.

The structure of the second higher Bruhat order relates to cubical tilings of zonotopes.

Abstract

The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is isomorphic to a poset on commutation classes of reduced words for the longest element of a type A Coxeter system. For this case, we prove that the noncontractible intervals are in natural correspondence with the zonogonal tilings of a zonogon. Our proof uses some tools developed by Felsner and Weil to show that the two standard orderings on the rhombic tilings of a zonogon are identical.