Polytopes with six vertices

Question

Let P(d,v) denote the number of convex polytopes (up to combinatorial equivalence) in $\,$d dimensions with v vertices. The octahedron and triangular prism are two familiar examples with d=3 and v=6 . There are five others bringing the total to P(3,6) = 7.

In four dimensions, any polytope with 6 vertices has just one more than the bare minimum needed to form a simplex. $\,$ Apparently there are P(4,6) = 4$\,$ such.

For completeness, there is only the hexagon if d=2 and only the simplex when d=5.$\,$ Therefore, the full count of all polytopes with v=6 and $2\le d$$\,$$\le 5$ $\,$is given by $\,$(1,7,4,1).

Questions: (1) Is there a reference which shows illustrations for the seven polytopes in the$\,$ d=3, v=6 case?

(2)$\,$The set of 3 x 3 doubly stochastic matrices forms a four dimensional polytope inside$\,$ $\Bbb R^9$$\,$having six vertices (corresponding to the six permutation matrices). Is there a nice way to embed this in$\,$$\,$$\Bbb R^4$$\,$ or to understand it in some more direct geometric way?

(3)$\,$Suppose we fix$\,$ v $\,$the number of vertices and consider the sequence P(d,v) for $\,$$2 \le d$$\,$$ \le v-1$. $\,$ Is this sequence known or conjectured to be unimodal?

Thanks very much

Answer 1

An answer to (2). Yes. Let $P=\|p_{ij}\|$ be a $3\times 3$ doubly stochastic matrix. The polytope belongs to the intersection of six hyperplanes:

$p_{11}+p_{12}+p_{13}=1$

$p_{21}+p_{22}+p_{23}=1$

$p_{31}+p_{32}+p_{33}=1$

$p_{11}+p_{21}+p_{31}=1$

$p_{12}+p_{22}+p_{32}=1$

$p_{13}+p_{23}+p_{33}=1$.

We can try to construct a basis in this intersection space and then describe the polytope by its coordinates with the given basis.

A more tricky but clear way is to consider a linear map $L$, $P\mapsto (p_{11},p_{13},p_{31},p_{33})$ . Since $P$ is a doubly stochastic matrix, the map $L$ is injective. Indeed, the elements of the matrix $P$ can be recovered from its image $L(P)$ as follows:

$p_{12}=1-p_{11}-p_{13}$,

$p_{32}=1-p_{31}-p_{33}$,

$p_{21}=1-p_{11}-p_{31}$,

$p_{23}=1-p_{13}-p_{33}$,

$p_{22}=1-p_{21}-p_{23}$.

The map $L$ maps the six permutation matrices to the points

$(1,0,0,1)$, $(1,0,0,0)$, $(0,0,0,1)$, $(0,0,1,0)$, $(0,1,0,0)$, and $(0,1,0,1)$.

Answer 2

Answer to (1): There is such a reference (by Professor Steven Dutch) showing Schlegel diagrams for nine types of polyhedra with six vertices (not seven); five under 'Polyhedra with 4 - 7 Faces' and four under 'Octahedra with 6 - 8 Vertices'.

Answer to (2): This is called the Birkhoff polytope $\Omega_3$. I believe this combinatorial type to be the same as the cyclic polytope C(6,4); also the direct sum of two triangles. It's straightforward to check that the two triangles with vertices at the even and odd permutations, respectively, aren't faces of any of the nine facets of $\Omega_3$. However, $\Omega_3$ is neighborly.