Kalai's squeezed 3-spheres are polytopal
Abstract
Description
In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal.
Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit.
11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometry
11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometry