Classifying two-faces of four-polytopes

Question

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to $\pi/2$), an the type of the polytope (all its vertices are proper, or all are ideal, or of mixed type), then one obtains some restrictions on the combinatorics. At the moment, I try to see if there are compact right-angled four-dimensional polytopes with volume bounded above by some (reasonably small) constant.

I have come to the following (probably, quite technical) question (or, better say, the following type of questions).

Question: Is there a convex simple polytope (in $\mathbb{R}^4$) which has $n=2,3,...,12$ hexagonal two-faces, while all other two-faces are pentagons.

Or, more generally, let $n_k \geq 0$ be the number of $k$-gonal faces ($k\geq 6$). Does there exist a convex simple polytope (in $\mathbb{R}^4$) which has given amounts $n_6$, $n_7$, ..., of, respectively, $6$-, $7$-, ...-gonal two-faces, while all other two-faces are pentagons?

I'm curious to know if there are general methods to study these (or similar) problems. Also, computational methods can be of interest (e.g. one can impose an upper bound on the number of three-facets of such a polytope and try to get a computer seek possible polytopes).

Thank you in advance for any help/references/etc!

Answer 1

I dont know the anser to the specific question. It seems that for the study of hyperbolic Coxeter polytopes even if using some properties of general simple 4-polytope one needs to use the very restricted nature of these Coxeter polytopes. Overall the question of prescribing the sizes of 2-faces of simple (and general) 4-polytopes (and d-polytope) is very interesting. For $d>4$ it is known that there is always a 2-face with three or four edges and this is easy for the simple case. For 4-polytopes it is conjectured (by Igor Pak) that if all 2-faces has at least 5 sides then the number of $k$-faces is at least that of the 120-cell. (This is quite delicate as it does not extend to duals of triangulations of homology spheres.)

The following answer https://mathoverflow.net/a/24340/1532 to Characterizing faces of 3-dimensional polyhedra. (Related to Victor Eberhard's Theorem [1890]:) fives some related results and questions.

To attack this question we can try to use The Dehn Sommerville relations to study the sequence $n_3,n_4,\dots, $ and to estimate the average number of sides in a polygon. This number is below 6 and it can be expressed in terms of two parameters of the dual polytope $g_1(P^*)$ (essentially the number of vertices) and $g_2(P^*)$. Understanding the sequences $(n_3(P),n_4(P),\dots )$ for 4-polytopes (and even very special classes like stacked polytopes) and in particular the cases where $n_3=n_4=0$ is indeed very interesting.