On the reconstruction of polytopes

Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon, David Yost

Published 2017-02-28Version 1

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 \emph{nonsimple} if the number of edges incident to it is $>d$. 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 $(d-3)$-skeleton and nonisomorphic face lattices. In particular, the result (1) is best possible for $4$-polytopes.