On the reconstruction of polytopes [article]

Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon, David Yost
2018 arXiv   pre-print
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
more » ... es with d-1 nonsimple vertices, isomorphic (d-3)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.
arXiv:1702.08739v4 fatcat:sihsekewqvcrtnyl4olxo4hmnq