## Classifying, realizing ...

### ... all combinatorial 3-spheres and 4-polytopes with up to 9 vertices

All combinatorial types are preceded by the flag f-vector $(f_0, f_1, f_2, f_3 ), f_{0,3}$.
 n=5 n=6 n=7 n=8 n=9 5<=n<=9 combinatorial 3-spheres 1 4 31 1336 316014 tar.gz 4-polytopes 1 4 31 1294 274148 tar.gz
all combinatorial 3-spheres, 9 vertices, with rational realization or non-realizability certificates
For n=9 vertices, for each combinatorial sphere, we provide either a realization or a certificate for non-realizability.
Each line contains a combinatorial sphere.
$(f_0, f_1, f_2, f_3 ), f_{0,3}$, [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:
1. this combinatorial type has a partial chirotope that contradicts Graßmann-Plücker: the partial chirotope and the GP-relation violated follows
2. this combinatorial type has a partial chirotope which can't be completed consistently. Relevant GP-relations and the contradiction follow
3. 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 vertices

Numbering 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.
 n=5 n=6 n=7 n=8 n=9 n=10 d=4 1 2 5 37 1142 162004 inscribed 1 2 5 37 1140 161978

### ... neighborly uniform oriented matroids

Numbering 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.
 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 d=4 1 1 1 3 23 431 13935 556061 d=5 1 1 2 126 159375 d=6 1 1 1 37 42099 d=7 1 1 4 35993
2-neighborly simplicial 6-polytopes with 10 vertices: 4523

### ... simplicial 3-spheres with small valence

Numbering corresponds to the enumeration of Frank Lutz. We provide rational realizations on the unit sphere.
4759 inscriptions of simplicial 3-spheres with small valence