On the Reconstruction of Polytopes

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d -polytope nonsimple if the number of edges incident to it is more than d . We show that (1) the face lattice of any d -polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d -polytope with at most d - 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d > 3 there are two d -polytopes with d - 1 nonsimple vertices, isomorphic ( d - 3 ) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.