Zonotopes whose cellular strings are all coherent

TL;DR

This paper classifies zonotopes with the all-coherence property, where all cellular strings are coherent, using their oriented matroid structure, and proves the invariance of this property.

Contribution

It provides a complete classification of all-coherent zonotopes based on their oriented matroid structure and establishes the invariance of the all-coherence property.

Findings

Complete classification of all-coherent zonotopes.

All-coherence property is invariant under oriented matroid structure.

Characterization of cellular strings in zonotopes.

Abstract

A cellular string of a polytope is a sequence of faces stacked on top of each other in a given direction. The poset of cellular strings, ordered by refinement, is known to be homotopy equivalent to a sphere. The subposet of coherent cellular strings is the face lattice of the fiber polytope, hence is homeomorphic to a sphere. In some special cases, every cellular string is coherent. Such polytopes are said to be all-coherent. We give a complete classification of zonotopes with the all-coherence property in terms of their oriented matroid structure. Although the face lattice of the fiber polytope in this case is not an oriented matroid invariant, we prove that the all-coherence property is invariant.