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12
Constructing neighborly polytopes and oriented matroids
"... Abstract. A dpolytope P is neighborly if every subset of ⌊ d ⌋ vertices is a face of P. In 1982, Shemer introduced 2 a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many dif ..."
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Abstract. A dpolytope P is neighborly if every subset of ⌊ d ⌋ vertices is a face of P. In 1982, Shemer introduced 2 a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer’s sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes. Résumé. Un dpolytope P est neighborly si tout sousensemble de ⌊ d ⌋ sommets forme une face de P. En 1982, She2 mer a introduit une construction de couture qui permet de rajouter un sommet à un polytope neighborly et d’obtenir
NONPROJECTABILITY OF POLYTOPE SKELETA
, 2009
"... We investigate necessary conditions for the existence of projections of polytopes that preserve full kskeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the giv ..."
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We investigate necessary conditions for the existence of projections of polytopes that preserve full kskeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to espace strictly preserves the kskeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of Rörig & Ziegler (2009) and complement their results.
Polyhedral Surfaces in Wedge Products
, 2007
"... We introduce the wedge product of two polytopes which is dual to the wreath product of Joswig and Lutz [6]. The wedge product of a pgon and a (q − 1)simplex contains many pgon faces of which we select a subcomplex corresponding to a surface. This surface is regular of type {p, 2q}, that is, all fa ..."
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We introduce the wedge product of two polytopes which is dual to the wreath product of Joswig and Lutz [6]. The wedge product of a pgon and a (q − 1)simplex contains many pgon faces of which we select a subcomplex corresponding to a surface. This surface is regular of type {p, 2q}, that is, all faces are pgons, all vertices have degree 2q, and the combinatorial automorphism group acts transitively on the flags of the surface. We show that for certain choices of parameters p and q there exists a realization of the wedge product such that the surface survives the projection to R 4. For a different choice of parameters such a realization does not exist.
PRODSIMPLICIALNEIGHBORLY POLYTOPES
, 2009
"... We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanya ..."
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We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products ” construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we moreover require the PSN polytope to be obtained as a projection of a polytope combinatorially equivalent to the product of r simplices, when the sum of their dimensions is at least 2k.
Polytopality and Cartesian products of graphs
 ACCEPTED IN ISRAEL JOURNAL OF MATHEMATICS
"... We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. ..."
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We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of nonpolytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (nonsimple) polytopal products whose factors are not polytopal.
<10.1007/s004540109311y>. <hal00777936>
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. PRODSIMPLICIALNEIGHBORLY POLYTOPES BENJAMIN MATSCHKE, JULIAN PFEIFLE, AND VINCENT PILAUD Abstract. Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their kskeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products ” construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k. 1.