Classifying, realizing ...
... all combinatorial 3-spheres and 4-polytopes with up to 9 verticesAll combinatorial types are preceded by the flag f-vector .
Each line contains a combinatorial sphere.
, [polytope|nonrealizable] type, additional data.
If the sphere is the boundary of a polytope, we provide rational coordinates of such a polytope.
Otherwise we provide a non-realizability certificate of a certain type. where "type" can be one of three values:
- this combinatorial type has a partial chirotope that contradicts Graßmann-Plücker: the partial chirotope and the GP-relation violated follows
- this combinatorial type has a partial chirotope which can't be completed consistently. Relevant GP-relations and the contradiction follow
- the completed chirotope admits a biquadratic final polynomial. We provide the completed chirotope and the linear program, which is infeasible
... and inscribing simplicial 3-spheres with up to 10 verticesNumbering corresponds to the enumeration of Frank Lutz. This is a table with the number simplicial 3-spheres with n vertices. Click on the file to download a file with rational realizations. Most of the realizations are inscribed in the unit sphere, the second row states how many of the realizations are inscribed.
... neighborly uniform oriented matroidsNumbering corresponds to the enumeration by Hiroyuki Miyata (宮田 洋行). Here is a table with the number of (simplicial) neighborly d-polytopes with n vertices for small d and n. Click on the number to download a file with rational realizations of these polytopes on the sphere. Some of the files are rather large.