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