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Can two non-equivalent polytopes of same dimension have the same graph?
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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?
graph-theory discrete-geometry convex-polytopes
asked Sep 4 at 11:31
M. Winter
554220
add a comment |Â
up vote
5
down vote
favorite
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?
graph-theory discrete-geometry convex-polytopes
asked Sep 4 at 11:31
M. Winter
554220
3
For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 at 12:54
In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 at 17:21
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
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?
graph-theory discrete-geometry convex-polytopes
asked Sep 4 at 11:31
M. Winter
554220
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?
graph-theory discrete-geometry convex-polytopes
asked Sep 4 at 11:31
M. Winter
554220
asked Sep 4 at 11:31
M. Winter
554220
asked Sep 4 at 11:31
M. Winter
554220
asked Sep 4 at 11:31
M. Winter
554220
554220
3
For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 at 12:54
In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 at 17:21
add a comment |Â
3
For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 at 12:54
In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 at 17:21
3
3
For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 at 12:54
For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 at 12:54
In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 at 17:21
In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 at 17:21
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
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.
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
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.
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
add a comment |Â
up vote
5
down vote
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.
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
add a comment |Â
up vote
5
down vote
up vote
5
down vote
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.
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
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.
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
answered Sep 4 at 12:12
Per Alexandersson
6,63973877
6,63973877
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
add a comment |Â
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you.
– M. Winter
Sep 4 at 12:16
add a comment |Â
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3
For $k=lfloor d/2rfloor$ your question is also answered on the wikipedia page en.wikipedia.org/wiki/Neighborly_polytope
– j.c.
Sep 4 at 12:54
In 3 dimensions, the 1-skeleton determines the combinatorial type. mathoverflow.net/a/308455/1345
– Ian Agol
Sep 5 at 17:21