The Unimodality Conjecture for cubical polytopes

TL;DR

This paper investigates the Unimodality Conjecture for cubical polytopes, proving it for dimensions less than 11 and providing a counterexample in 12 dimensions, highlighting the conjecture's limitations.

Contribution

It proves the Unimodality Conjecture for cubical polytopes in dimensions under 11 and constructs a 12-dimensional counterexample, advancing understanding of face vector properties.

Findings

Unimodality holds for dimensions less than 11.

Counterexample in 12 dimensions with non-unimodal face vector.

First third of face vector is increasing, last third decreasing.

Abstract

Although the Unimodality Conjecture holds for some certain classes of cubical polytopes (e.g. cubes, capped cubical polytopes, neighborly cubical polytopes), it fails for cubical polytopes in general. A 12-dimensional cubical polytope with non-unimodal face vector is constructed by using capping operations over a neighborly cubical polytope with 2 to the power 131 vertices. For cubical polytopes, the Unimodality Conjecture is proved for dimensions less than 11. The first one-third of the face vector of a cubical polytope is increasing and its last one-third is decreasing in any dimension.