The smallest non-trivial 2-simplicial and 2-simple 4-polytope.

A polytope P is *k-simplicial* if all its k-faces are simplices and it is *h-simple* if its dual polytope P^{Δ} is h-simplicial [2].
Therefore, a 4-dimensional polytope is 2-simplicial and 2-simple, if all 2-dimensional faces are triangles and all edges are contained in exactly 3 facets;
in this case, we call it a *2s2s-polytope*. Trivially, the 4-simplex is a 2s2s-polytope.

The flag vectors of 2s2s-polytopes satisfy two linear inequalities, valid for all flag vectors of 4-polytopes, with equality and hence are contained in a 2-dimensional face of the flag vector cone of 4-polytopes [1]. It was first shown by Paffenholz and Ziegler [5] that there exist infinitely many 2s2s-polytopes on an arbitrarily large number of vertices.

It is then natural to ask for the existence of 2s2s-polytopes on few vertices.
The polytope W_{9} on 9 vertices, of which a Schlegel diagram is given in the first figure,
is the combinatorially unique smallest non-trivial 2s2s-polytope.

This can be proved by elementary methods in several steps. Let P be a 2s2s-polytope with at most 9 vertices that is not the simplex; then one can show:

- P cannot have less than 9 vertices.
- P contains at most one facet with 6 vertices.
- if P contains an octahedron then it is combinatorially equivalent to W
_{9}. - P or P
^{Δ}contains an octahedron. - W
_{9}is self-dual.

The polytope given here generalises to a whole family of 2s2s-polytopes with a further, very interesting property.
A d-polytope is *elementary* if it satisfies g_{2}=0, where

g_{2}= f_{02}- 3 f_{2}+ f_{1}- d f_{0}+ d(d+1)/2

with the (reduced) flag vector (f_{0},f_{1},f_{2};f_{02}).
Kalai [3] showed that g_{2}>=0 is a valid linear inequality for all flag vectors of 4-polytopes.
Elementary 2s2s-polytopes can therefore be found on a ray l_{1} of the flag vector cone (see [1]).

Until recently, only two flag vectors of polytopes were known to lie on l_{1}. The polytope W_{9} is elementary,
which can be easily calculated from its f-vector (9,26,26,9) and from the fact that it is 2-simplicial (i.e. f_{02}=3f_{2}).
By performing a sequence of "pseudo-stacking steps", that is, placing new vertices beyond certain facets in a suitable way,
one can construct elementary 2s2s-polytopes W_{4k+1} on 4k+1 vertices for all k>=1.
The construction is described in detail in [4], together with other examples that cover all vertex numbers n>=9, n≠12.

The second figure shows a different Schlegel diagram of W_{9}.
Pseudo-stacking starts beyond the simplex facet, whose vertices are coloured red in all the figures, and adds four vertices.
This yields the next example, W_{13}, depicted in the third figure, with new edges arising in the pseudo-stacking process hilighted.

Keywords | Polytope; Discrete Geometry | |

MSC-2000 Classification | 52B05 | |

Zentralblatt No. | 05264906 |

- Bayer, Margaret M.:
*The Extended f-Vectors of 4-Polytopes*, J. Combin. Theory Ser. A**44**(1987), 141--151. - Grünbaum, Branko: Convex Polytopes, 2nd edition, Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler, Springer-Verlag (2003).
- Kalai, Gil:
*Rigidity and the lower bound theorem. I*, Invent. Math.**88**(1987), 125--151. - Paffenholz, Andreas and Werner, Axel:
*Constructions for 4-Polytopes and the Cone of Flag Vectors*, Cont. Math.**423**(2007), 283--303, http://de.arxiv.org/abs/math.CO/0511751. - Paffenholz, Andreas and Ziegler, Günter M.:
*The E-construction for lattices, spheres and polytopes*, Discrete Comput. Geom.**32**(2004), 601--621.

- Master File: W9.poly
- Applet File: W9_symm.jvx
- Applet File: W9.jvx
- Applet File: W13.jvx
- Preview: W9_symm.jpg
- Preview: W9.jpg
- Preview: W13.jpg

Submitted: Wed Dec 14 15:51:58 CET 2005.

Revised: Mon Mar 19 16:51:00 CET 2007.

Accepted: Tue Jun 5 12:50:52 CEST 2007.

TU Berlin

Inst. f. Mathematik, MA 6-2

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Germany

awerner@math.tu-berlin.de