Cubical 4-Polytopes with Few Vertices

A cubical polytope is a convex polytope all of whose facets are conbinatorial cubes. A d-polytope Pis called almost simple if, in the graph of P, each vertex of Pis d-valent of (d+ 1)-valent. It is known that, for d> 4, all but one cubical d-polytopes with up to 2^d+1vertices are almost simple, which provides a complete enumeration of all the cubical d-polytopes with up to 2^d+1vertices. We show that this result is also true for d=4.