# On the reconstruction of polytopes

## TL;DR

This paper investigates how the face lattice of polytopes can be reconstructed from their skeletons, establishing new bounds based on the number of nonsimple vertices and showing limitations for higher counts.

## Contribution

It extends previous results by determining face lattices from skeletons for polytopes with limited nonsimple vertices and identifies cases where reconstruction is impossible.

## Key findings

- Face lattice determined by 1-skeleton with up to two nonsimple vertices.
- Face lattice determined by 2-skeleton with up to d-2 nonsimple vertices.
- Existence of nonisomorphic polytopes with high nonsimple vertices sharing lower-dimensional skeleta.

## 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.