Realizations of the associahedron and cyclohedron
type: Item
instance of: scholarly article
title: Realizations of the associahedron and cyclohedron
zbMATH Open document ID: 1125.52011
author: Christophe Hohlweg
author: Carsten E. M. C. Lange
published in: Discrete \& Computational Geometry
publication date: 2007-06-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0510614
review text: The associahedron first arose purely as the graph corresponding to the bistellar operations on triangulations of a convex polygon which only use the vertices of the polygon. Later, it was shown how to realize this graph as the edge-graph of a suitable simple convex polytope. Indeed, this polytope can be taken to be the secondary polytope (a special case of a fibre polytope) of the projection of an appropriate simplex on the polygon. In a similar way, the cyclohedron corresponds to centrally symmetric triangulations of a centrally symmetric polygon. There is also a connexion with the permutahedra arising from the Coxeter groups of types \(A\) and \(B\), respectively. In this paper, the authors discuss realizations of these polytopes with integer coordinates for their vertices, comparing them with these permutahedra; these coordinates are obtained through an algorithm which uses the corresponding oriented Coxeter graphs of types \(A\) and \(B\) as the only input data.
Mathematics Subject Classification ID: 52B12
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 5166264
zbMATH Keywords: associahedron
zbMATH Keywords: permutahedron
zbMATH Keywords: cyclohedron
zbMATH Keywords: Coxeter graph
reviewed by: Peter McMullen
MaRDI profile type: Publication
OpenAlex ID: W2019019197
arXiv ID: math/0510614
DOI: 10.1007/S00454-007-1319-6
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description: scientific article