Can two non-equivalent polytopes of same dimension have the same graph?

By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. their face lattices are isomorphic.

I know that two polytopes can have isomorphic graphs while being non-equivalent, e.g. neighborly polytopes. However, all examples I know of are polytopes of different dimension. So I wonder:

Question: Can there be two non-equivalent poyltopes of the same dimension with the same graph?

Especially, are all $k$-neighborly polytopes of the same dimension equivalent?

For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 '18 at 12:54

In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 '18 at 17:21

1 Answer
1

The 1-skeleton is usually not enough to recover the face lattice,
but under some conditions it is.
I did a quick google search, and read the abstract in this paper.

I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 '18 at 12:16

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