  We show that there are simple 4-dimensional polytopes with n vertices such
that all separators of the graph have size at least $\Omega(n/\log n)$. This
establishes a strong form of a claim by Thurston, for which the construction
and proof had been lost.
  We construct the polytopes by cutting off the vertices and then the edges of
a particular type of neighborly cubical polytopes. The graphs of simple
polytopes thus obtained are 4-regular; they contain 3-regular "cube-connected
cycle graphs" as minors of spanning subgraphs.
