Timeline for An example of a "simple poset" which does not belong to a convex polytope
Current License: CC BY-SA 4.0
27 events
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| Oct 1, 2019 at 11:39 | comment | added | David E Speyer | Yes, your understanding of the (8, 28, 40, 20) data is right. I am confused by the same issue @SamHopkins is confused by. The face poset on the webpage I linked is not realizable. I suspect that its edge graph already is already not realizable. It's $f$-vector certainly IS realizable. Which of these am I supposed to be considering. | |
| Sep 30, 2019 at 3:26 | comment | added | Sam Hopkins | It's true that the $g$-conjecture/theorem is quite an amazing thing, but I'm trying to stress again that the "combinatorial structure" of a polytope is much more than its $f$-vector. | |
| Sep 30, 2019 at 1:34 | comment | added | giulio bullsaver | anyway if I understand it correctly, this g-theorem is basically the answer to my question: given the data (F,V) I can get the g-vector and check McMullen's conditions. That's enough to establish whether (F,V) arise or not from a simple/simplicial convex polytope...wow | |
| Sep 30, 2019 at 1:20 | comment | added | Sam Hopkins | @giuliobullsaver: Actually, to amend my previous comment- the $g$-conjecture (see en.wikipedia.org/wiki/Simplicial_sphere) for simplicial spheres says that the set of $f$-vectors realizable by simplicial spheres should be the same as those realizable by simplicial polytopes. | |
| Sep 30, 2019 at 1:19 | comment | added | Sam Hopkins | @giuliobullsaver: That's (probably) true (I didn't check it); but the $f$-vector being realizable is not the same as the whole simplicial complex being realizable. | |
| Sep 30, 2019 at 1:12 | comment | added | giulio bullsaver | I have been studying more about convex realizability of simple (or simplicial polytopes), and it seems that the f-vector (8,28,40,20) satisfies McMullen's condition and thus should correspond to a convex polytope. Am I right? | |
| Sep 29, 2019 at 21:25 | comment | added | Sam Hopkins | (where I wrote "vertices and edges" above I of course meant "vertices and facets") | |
| Sep 29, 2019 at 20:35 | comment | added | giulio bullsaver | @David Am I right in understanding that I can think of the data defining (8, 28, 40, 20) non-real as being the list of vertices, each represented as the intersection of four facets? The corresponding simple "polytope" is proven to be non-realizable as a convex polytope? | |
| Sep 29, 2019 at 20:32 | vote | accept | giulio bullsaver | ||
| Sep 29, 2019 at 18:06 | comment | added | giulio bullsaver | yes I meant facet | |
| Sep 29, 2019 at 18:03 | comment | added | giulio bullsaver | oooh I see what you mean. The examples of that link, once dualized should give an example of my question. | |
| Sep 29, 2019 at 18:01 | comment | added | M. Winter | @giuliobullsaver Do you mean the vertex appears in 10 facets? If so, I suppose Sam is right, and you should dualize first, as I assume the f-vectors you stated belongs to a simplicial sphere. | |
| Sep 29, 2019 at 17:50 | comment | added | giulio bullsaver | Dear Sam, no I am really interested in simple polytopes not simplicial ones. Although I guess that the difference is somewhat unimportant because of the duality... | |
| Sep 29, 2019 at 17:35 | comment | added | Sam Hopkins | @giuliobullsaver: I think you may need to dualize (swapping the roles of vertices and edges) as simplicial spheres are a generalization of simplicial polytopes, not simple polytopes. | |
| Sep 29, 2019 at 14:58 | comment | added | giulio bullsaver | Dear David, if I understand correctly the notation of those data, both non-real entries (8,27,38,19) and (8,28,40,20) cannot correspond to a "simple poset" because the vertex "1" appears on ten faces. | |
| Sep 29, 2019 at 13:15 | comment | added | giulio bullsaver | thank you David, I will do it. In the meanwhile I am updating my question to make clearer what I was looking for | |
| Sep 29, 2019 at 13:13 | comment | added | David E Speyer | More specifically, search for "(8, 27, 38, 19) 76 non-real" and "(8, 28, 40, 20) 80 non-real" to find non-polytopal simplicial examples. | |
| Sep 29, 2019 at 13:08 | comment | added | David E Speyer | Go to ftp.imp.fu-berlin.de/pub/moritz/classification/… and search for the word "non-real". | |
| Sep 29, 2019 at 12:23 | comment | added | giulio bullsaver | Dear Winter, the examples you provided - although interesting - are not what I was looking for: just by looking at the combinatorics of these posets you can prove that they can't be the face lattice of a convex polytope. In the hemicube this is so because the 3 putative facets meet at more than three vertices (therefore their hyperplanes would be the same). For the hemidodecahedron the poset is non-orientable (start at a vertex and choose an orientation of the 3 facets intersecting there, travel around vertices 1 through 9 in wikipedia's picture and get back at 1 with different orientation) | |
| Sep 29, 2019 at 11:58 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Sep 29, 2019 at 10:41 | comment | added | M. Winter | @giuliobullsaver I added an explicit example you might like: the hemi dodecahedron. | |
| Sep 29, 2019 at 10:37 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Sep 29, 2019 at 10:21 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Sep 28, 2019 at 21:47 | comment | added | M. Winter | @giuliobullsaver Right now, I cannot. But Wikipedia states that Grünbaum explicitly constructed such a simplicial sphere (the smallest example known for $d=4$ has eight vertices). Unfortunately no source is given for that claim. I might err, but I remember to have read something like this in Ziegler's "Lectures on Polytopes" (I might check later). But I assume it should not be too hard to track down this claim. | |
| Sep 28, 2019 at 20:45 | comment | added | giulio bullsaver | Every time I asked this question to friend mathematicians I eventually get this answer. But would you be able to provide an explicit example? That is to give all combinatorical data I need to define a poset which is not the face lattice of a convex polytope? | |
| Sep 28, 2019 at 20:33 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Sep 28, 2019 at 20:14 | history | answered | M. Winter | CC BY-SA 4.0 |