Amos Altshuler

The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes

The classification of the 1296 (simplicial) 3-spheres with nine vertices into polytopal and nonpolytopal spheres, started earlier, is completed here. It is shown that there are 1142 polytopal and 154 nonpolytopal such spheres, and a fast procedure for their construction is described.

An enumeration of combinatorial 3-manifolds with nine vertices

Abstract A complete classification is given for non-neighborly combinatorial 3-manifolds with nine vertices. It is found that there are 1246 such types, and that they all are spheres. It is shown that 1057 of those spheres are polytopal. i.e. can be realized as boundary complexes of convex 4-polytopes. 115 spheres are non-polytopal, and 74 spheres remain undecided.

Neighborly 2-Manifolds with 12 Vertices

We provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space. Whereas the question of embeddability remains open in its general form, we obtain several properties of the embedding (polyhedral realization) under the assumption that it does exist: 1.The order of the geometrical automorphism group of any polyhedral realization would not exceed 2. 2.The polyhedral realization would not be obtainable via the Schlegel diagram of any 4-polytope; moreover, none of our orientable neighborly 2-manifolds with 12 vertices can be found within of the 2-skeleton of any 4-polytope. 3.The polyhedral realization would not allow a tetrahedral subdivision without inserting new vertices. By using a weaker version of the manifold property, we obtain neighborly polyhedra with 2nvertices for everyn?3.

Combinatorial 3-manifolds with few vertices

Abstract It is proved that every combinatorial 3-manifold with at most eight vertices is a combinatorial sphere.

Construction theorems for polytopes

Certain construction theorems are represented, which facilitate an inductive combinatorial construction of polytopes. That is, applying the constructions to ad-polytope withn vertices, given combinatorially, one gets many combinatoriald-polytopes—and polytopes only—withn+1 vertices. The constructions are strong enough to yield from the 4-simplex all the 1330 4-polytopes with up to 8 vertices.

Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.

Neighborly maps with few vertices

A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>−10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.

Construction and Representation of Neighborly Manifolds

We deal with neighborly 2-manifolds and 2-pseudomanifolds. We describe a representation of these objects, which is independent of the particular labelling of their vertices. This representation also enables an easy calculation of the automorphism group. We describe two operations which, when applied to one such object, enable us to obtain many (and perhaps all) other such objects with the same number of vertices. Finally, we apply those operations to manifolds and pseudomanifolds withvvertices, for some small values ofv.

On weakly neighborly polyhedral maps of arbitrary genus

A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We lay the foundation for an investigation of the w.n.p. maps of arbitrary genus. In particular we find all the w.n.p. maps of genus 0, we prove that for everyg> the number of w.n.p. maps of genusg (orientable or not) is finite, and we find an upper bound for the number of vertices in a w.n.p. map of genusg>0. This upper bound grows as (4g(2/3) wheng→∞.

The weakly neighborly polyhedral maps on the torus

A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without a boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just five distinct w.n.p. maps on the torus, and that only three of them are geometrically realizable as polyhedra with convex faces.