Cubical \(d\)-polytopes with few vertices for \(d>4\) (Q1360275)

A cubical polytope is a convex polytope all of whose facets are combinatorial cubes. In this paper, the authors obtain a complete enumeration of the cubical d-polytopes with less than \(2^{d+1}\) vertices for \(d > 4\), by showing that all but one of these polytopes are almost simple, that is to say, each of their vertices has either valency \(d\) or \(d+1\). The authors point out that this result is also valid for \(d = 4\) but is not true for \(d = 3\).