Locally projective polytopes of type

This paper attempts to classify the locally projective section regular n-polytopes of type {4,3,…,3,p}, that is, to classify polytopes whose facets are cubes or hemicubes, and the vertex figures are spherical or projective polytopes of type {3,…,3,p}, with the facets and vertex figures being not both spherical. Spherical or projective (n−1)-polytopes of type {3,…,3,p} only exist when p4, or p=5 and n−14, or n−1=2. However, some existence and non-existence results are obtained for other values of p and n. In particular, a link is derived between the existence of polytopes of certain types, and vertex-colourability of certain graphs. The main result of the paper is that locally projective section regular n-polytopes exist only when p=4, or when p=5 and n=4 or 5.     

Über die Symmetriegruppen von regulärseitigen Polytopen

Convex polytopes are called regular faced, if all their facets are regular. It is known, that all regular faced 3-polytopes have a nontrivial symmetry group, and also alld-polytopes with centrally symmetric facets. Here it is shown, that there ecist in fact regular facedd-polytopes with trivial symmetry group, but only ford=4. The corresponding class of polytopes is studied.     

Twisted Groups and Locally Toroidal Regular Polytopes

In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks and . For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

     

Lattice-free polytopes and their diameter

A convex polytope in real Euclidean space islattice-free if it intersects some lattice in space exactly in its vertex set. Lattice-free polytopes form a large and computationally hard class, and arise in many combinatorial and algorithmic contexts. In this article, affine and combinatorial properties of such polytopes are studied. First, bounds on some invariants, such as the diameter and layer-number, are given. It is shown that the diameter of ad-dimensional lattice-free polytope isO(d ³). A bound ofO(nd+d ³) on the diameter of ad-polytope withn facets is deduced for a large class of integer polytopes. Second, Delaunay polytopes and [0, 1]-polytopes, which form major subclasses of lattice-free polytopes, are considered. It is shown that, up to affine equivalence, for anyd≥3 there are infinitely manyd-dimensional lattice-free polytopes but only finitely many Delaunay and [0, 1]-polytopes. Combinatorial-types of lattice-free polytopes are discussed, and the inclusion relations among the subclasses above are examined. It is shown that the classes of combinatorial-types of Delaunay polytopes and [0,1]-polytopes are mutually incomparable starting in dimension six, and that both are strictly contained in the class of combinatorial-types of all lattice-free polytopes. This research was supported by DIMACS—the Center for Discrete Mathematics and Theoretical Computer Science at Rutgers University.     

Regular polytopes of type {4,4,3} and {4,4,4}

Abstract regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schläfli types {4,4,3} and {4,4,4}.     

Hamiltonian Submanifolds of Regular Polytopes

We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k -Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S ²×S ². By this example all regular cases of n vertices with n<20 or, equivalently, all cases of regular d-polytopes with d≤9 are now decided.     

Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88-91] and the second author [P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n+3 facets, Electron. J. Combin. 14 (2007), R69, 36 pp.], this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.     

Flat regular polytopes

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP ₁ andP ₂, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP ₁ and whose vertex-figures are isomorphic toP ₂? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.     

Polytopes which are orthogonal projections of regular simplexes

We consider the polytopes which are certain orthogonal projections of k-dimensional regular simplexes in k-dimensional Euclidean space R ^k . We call such polytopes -polytopes. Every sufficiently symmetric polytope, such as a regular polytope, a quasi-regular polyhedron, etc., belongs to this class. We denote by P _m,n all n-dimensional -polytopes with m vertices. We show that there is a one-to-one correspondence between the elements of P _m,n and those of P _m,m–n–1 and that this correspondence preserves the symmetry of -polytopes. Using this duality, we determine some of the P _m,n 's. We also show that a -polytope is an orthogonal projection of a cross polytope if and only if it has central symmetry.     

Polytopes of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5?n?9 are available. Two observations arise when we look at the results: (1) for n?5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n?7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n?7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3?r?n−1.     

Simpler Tests for Semisparse Subgroups

The main results of this article facilitate the search for quotients of regular abstract polytopes. A common approach in the study of abstract polytopes is to construct polytopes with specified facets and vertex figures. Any nonregular polytope may be constructed as a quotient of a regular polytope by a (so-called) semisparse subgroup of its automorphism group W (which will be a string C-group). It becomes important, therefore, to be able to identify whether or not a given subgroup N of a string C-group W is semisparse. This article proves a number of properties of semisparse subgroups. These properties may be used to test for semisparseness in a way which is computationally more efficient than previous methods. The methods are used to find an example of a section regular polytope of type {6, 3, 3} whose facets are Klein bottles. Received February 15, 2005     

A lower bound on the number of sharp shadow-boundaries of convex polytopes

We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812.     

Locally toroidal regular polytopes of rank 4

The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3,p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with the same Schläfli-symbol in hyperbolic 3-space. Also the paper discusses regular tessellations (secretions of rank 3) which are locally inscribed into regular 4-polytopes. In particular, this leads to local criteria for the finiteness of the polytopes.     

Toric Ideals and an Infinite Family of Normal (0, 1)-Polytopes without Unimodular Regular Triangulations

Abstract. An infinite family of normal (0,1) -polytopes, none of whose unimodular triangulations is regular, is constructed. For the construction, we discuss triangulations of convex polytopes and flips of triangulations by studying markings on the set of circuits of toric ideals.     

Revlex-initial 0/1-polytopes

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers g_nfac(d,n) of facets and the minimum average degree g_avdeg(d,n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy g_nfac(d,n)?3d and g_avdeg(d,n)?d+4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.     

On the Volume Formulas of Cones and Orthogonal Multi-cones in S^n（1） and H^n（-1）

Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sⁿ(1) and Hⁿ(—1). (Dedicated to the memory of Shiing-Shen Chern)     

Locally toroidal polytopes and modular linear groups

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d≥2, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, here we develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.     

Colorful polytopes and graphs

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P _G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P _G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.