Combinatorial types of bicyclic polytopes

We classify the bicyclic polytopes and their vertex figures, up to combinatorial equivalence. These four-dimensional polytopes, which were previously studied by Smilansky, admit abelian groups of orientation-preserving symmetries that act transitively on their vertices. The bicyclic polytopes come in both simplicial and nonsimplicial varieties. It is noteworthy that their facial structures admit an explicit and complete presentation. Their vertex figures are also of interest, and they play a prominent role in the classification; their combinatorial structures are studied in detail here. The ff-vectors of the polytopes and of their vertex figures are given.