Kalai's squeezed 3-spheres are polytopal

In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit.Comment: 11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometr

Separation index of graphs and stacked 2-spheres

In 1987, Kalai proved that stacked spheres of dimension $d\geq 3$ are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension $d=2$. In this article, we give a characterisation of stacked $2$-spheres using what we call the {\em separation index}. Namely, we show that the separation index of a triangulated $2$-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all $n$-vertex triangulated $2$-spheres, the separation index is {\em minimised} by some $n$-vertex flag sphere for $n\geq 6$. Furthermore, we apply this characterisation of stacked $2$-spheres to settle the outstanding $3$-dimensional case of the Lutz-Sulanke-Swartz conjecture that "tight-neighbourly triangulated manifolds are tight". For dimension $d\geq 4$, the conjecture has already been proved by Effenberger following a result of Novik and Swartz.Comment: Some typos corrected, to appear in "Journal of Combinatorial Theory A

Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres

The aim of this paper is to give a survey of the known results concerning centrally symmetric polytopes, spheres, and manifolds. We further enumerate nearly neighborly centrally symmetric spheres and centrally symmetric products of spheres with dihedral or cyclic symmetry on few vertices, and we present an infinite series of vertex-transitive nearly neighborly centrally symmetric 3-spheres.Comment: 26 pages, 8 figure

Combinatorial 3-manifolds with 10 vertices

We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product $S^2\times S^1$ and 615 triangulations of the twisted sphere product S^2_\times_S^1. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertex-minimal non-shellable 3-balls with 9 vertices.Comment: 9 pages, minor revisions, to appear in Beitr. Algebra Geo

Tight triangulations of closed 3-manifolds

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $\mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $\mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $\mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $\mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $\beta_1$, then $(n-4)(617n- 3861) \leq 15444\beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.Comment: 21 pages, 1 figur