Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep track of their provenance. The nice thing about the simplicial case is that in order to get the whole combinatorial structure all one needs is the list of the facets. My question:

Are there any "comprehensive" resources for examples of simplicial polytopes or spheres?

Here by comprehensive I mean either examples obtained via some construction method or an aggregate of various results in the field. These examples could come in any form I guess, but a simple list of facets would probably be the easiest to work with.

I've been searching around a bit and found the following sources:

  • John Palmieri gives an implementation in Sage of a few interesting examples of nonshellable spheres and manifolds here.

  • Frank Lutz gives examples of small non-constructible spheres here and he also has a few papers enumerating $3$-manifolds with $10$ vertices here and vertex-transitive triangulations (with E. Köhler) here. Although in the latter $2$ papers the facet-lists are not given.

I am sure there are many other papers that give examples of various interesting triangulations that I am not aware of so even if you do not know of a database of such examples, answering with specific papers would be useful, I think.

Thank you!

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    $\begingroup$ Just to clarify: I did add some examples to Sage, but not the nonshellable spheres. $\endgroup$ – John Palmieri Sep 14 '15 at 22:14

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:

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)

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    $\begingroup$ Thank you, these are great resources. I also found this library of triangulations especially useful. $\endgroup$ – Alexandru Papiu Sep 15 '15 at 22:20

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

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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.

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