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arXiv · 1702.08739

On the reconstruction of polytopes

Joseph Doolittle·Eran Nevo·Guillermo Pineda-Villavicencio·Julien Ugon·David Yost
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Abstract

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 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.

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Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon, David Yost. 2017-02-28. On the reconstruction of polytopes. https://arxiv.org/abs/1702.08739

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