arXiv subjects
Publications and source records attributed to Andreas Paffenholz.
The toric ideals of $3\times 3$ transportation polytopes $T$ are quadratically generated. The only exception is the Birkhoff polytope $B_3$. If $T$ is not a multiple of $B_3$, these ideals even have squarefree quadratic initial ideals. This class contains all smooth $3 \times 3$ transportation polytopes.
We construct a new 2-parameter family E_mn of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E_44 is the 24-cell. For large m,n the f-vectors have ``fatness'' close to 6. The E_t-construction of Paffenholz and Ziegler applied to products of polygons yields cellular spheres with the combinatorial structure of E_mn. Here we prove polytopality of these spheres. More generally, we construct polytopal realisations for spheres obtained from the E_t-construction applied to products of polytopes in any dimension d>=3, if these polytopes satisfy some consistency conditions. We show that the projective realisation space of E_33 is at least nine dimensional and that of E_44 at least four dimensional. This proves that the 24-cell is not projectively unique. All E_mn for relatively prime m,n>= 5 have automorphisms of their face lattice not induced by an affine transformation of any geometric realisation. The group Z_m x Z_n generated by rotations in the two polygons is a subgroup of the automorphisms of the face lattice of E_mn. However, there are only five pairs (m,n) for which this subgroup is geometrically realisable.
In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that "cut across an ideal." Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P,I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular "generalized Bier spheres." In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields "many shellable spheres", most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.
We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct for each $d\ge3$ an infinite family of $(d-2)$-simplicial 2-simple $d$-polytopes, thus solving a problem of Grünbaum.