polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tu-berlin.de/diskregeom/polymake/.

Steinitz (1906) gave a remarkably simple and explicit description of the set of all f-vectors f(P ) = (f0 , f1 , f2) of all 3-dimensional convex polytopes. His result also identifies the simple and the simplicial 3-dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2-spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).

INTRODUCTION The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs--- subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t

A well-known theorem of Blind and Mani [4] says that every simple polytope is uniquely determined by its graph. In [11] Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai’s proof can be suitably generalized without much effort. We apply our results to a special class of cubical polytopes.

It is an open problem to characterize the cone of f-vectors of 4-dimensional convex polytopes. The question whether the “fatness ” of the fvector of a 4-polytope can be arbitrarily large is a key problem in this context. Here we construct a 2-parameter family of 4-dimensional polytopes π(P 2r n) with extreme combinatorial structure. In this family, the “fatness ” of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the “complexity, ” gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to R4.

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Abstract. We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g2 = 0 on any number of n ≥ 13 vertices. In particular, this implies that the ray ℓ1, described by Bayer (1987), is fully contained in the convex hull of all flag vectors of 4-polytopes. Especially interesting examples on 9, 10 and 11 vertices are presented. 1.

The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background. 1

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Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills [16] can be embedded into 3. 1.