"Database" of simplicial polytopes/spheres

You can find Frank Lutz's lists of simplicial spheres (and other manifolds) here:

Let me shamelessly self-advertise my list of simplicial 4-polytopes with up to $10$ vertices and various families of neighborly polytopes here. I give realization with rational coordinates (inscribed, if possible). You can extract the lists of facets easily from the coordinates. For the simplicial 4-polytopes with 10 vertices I use the same numbering as Frank Lutz uses for the simplicial 3-spheres. Of course some of the spheres are non-realizable ($85\ 878$ to be precise), and then the corresponding number does not appear in my list.

There is an arxiv preprint summarizing the results: Realizability and inscribability for some simplicial spheres and matroid polytopes.

You might also be interested in the following pages by Hiroyuki Miyata:

• http://www-imai.is.s.u-tokyo.ac.jp/~hmiyata/oriented_matroids/
• https://sites.google.com/site/hmiyata1984/neighborly_polytopes

and Lukas Finschi's "Homepage of Oriented Matroids".

For (simplicial) 3-dimensional polytopes, it is very very fast to generate the combinatorial types of triangulations using plantri, by Gunnar Brinkmann and Brendan McKay, so there really is no need for a database. (At least for polytopes with a small number of vertices. A database of "interesting polytopes" and not "all polytopes up to a certian number of vertices" would still be something nice to have)

For some early very related classifications (ordered chronologically) see

1. Neighborly 4-polytopes with 9 vertices Altshuler and Steinberg - J Comb Theory, Series A, 1973‏

2. An enumeration of combinatorial 3-manifolds with nine vertices Altshuler and Steinberg, Discrete Math, 1976

3. Neighborly 4-polytopes and neighborly combinatorial 3-manifolds with ten vertices, Altshuler, Canad J Math 1977

4. The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes, Altshuler, Bokowski and Steinberg, Discrete Math, 1980

5. The complete enumeration of the 4-polytopes and 3-spheres with eight vertices, Altshuler and Steinberg, Pacific J Math, 1985‏

6. Neighborly 2-manifolds with 12 vertices Altshuler, Bokowski and Schuchert, J Comb Theory, Series A, 1996‏

I want to mention a couple of resources that I haven't seen anyone else put up:

• The simpcomp GAP package includes a simplicial complex library. As I understand it, it includes Frank Lutz's list (and some more stuff). See the simpcomp documentation.

• Masahiro Hachimori has a small library of simplicial complexes (really more of an encyclopedia) with various properties. It's well-curated (if not updated since 2001) and a good place to start looking for some standard-ish examples. I've found it a useful place to start looking on several occasions.

• On a different note, Frank Lutz started a journal-type collection of Electronic Geometry Models. The emphasis is more on visualization than facet lists. If you want to e.g. visualize Rudin's non-shellable ball, though, then this is the place to go! Unfortunately, it's looking a bit dated: the last submission was 2013, and the visualization requires java for best results.