Theory of families of polytopes: fullerenes and Pogorelov polytopes
type: Item
instance of: scholarly article
title: Theory of families of polytopes: fullerenes and Pogorelov polytopes
zbMATH Open document ID: 1473.52020
DOI: 10.3103/S0027132221020042
published in: Moscow University Mathematics Bulletin
publication date: 2021-09-08T00:00:00Z
review text: This is a review on older and recent results (in part by the author) on certain families of three-dimensional polytopes. They can be defined via the notion of \(k\)-belt. This is a cyclic sequence of \(k\) faces of the given polytope in which only successive faces are adjacent and no three faces have a common vertex. The author defines five classes of simple \(3\)-polytopes by existence, non-existence, or triviality (defined by surrounding a face) of \(k\)-belts for various \(k\). Among them are flag polytopes and Pogorelov polytopes. These polytope classes play an important role in hyperbolic geometry and algebraic topology, but also in chemistry, namely the fullerenes. The latter have only \(5\)-gonal and \(6\)-gonal faces, and an \(n\)-disk fullerene, which plays a role here, has in addition one \(n\)-gonal face with \(n\not=5,6\). The author reviews properties and characterizations of these classes of polytopes, their role in hyperbolic geometry, constructions to obtain such polytopes from simpler ones, and the construction of Fullerenes using growth operations. The last two sections provide a comprehensive survey of the use of the considered polytopes, in particular Pogorelov polytopes, in the construction of cohomologically rigid families of manifolds and in Thurston's geometrization of three-dimensional manifolds.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 52B05
zbMATH DE Number: 7392401
zbMATH Keywords: three-dimensional polytope
zbMATH Keywords: belt of faces
zbMATH Keywords: cyclic \(k\)-edge connectivity
zbMATH Keywords: fullerene
zbMATH Keywords: Pogorelov polytope
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: cohomological rigidity
zbMATH Keywords: Thurston geometrization
author: N. Yu. Erokhovets
MaRDI profile type: Publication
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full work available at URL: s0027132221020042
OpenAlex ID: W3186941465
description: scientific article; zbMATH DE number 7392401