Construction of fullerenes and Pogorelov polytopes with 5-, 6- and one 7-gonal face
type: Item
instance of: scholarly article
title: Construction of fullerenes and Pogorelov polytopes with 5-, 6- and one 7-gonal face
zbMATH Open document ID: 1392.52006
author: N. Yu. Erokhovets
published in: Symmetry
publication date: 2018-08-13T00:00:00Z
review text: Summary: A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed numbers of \(k\)-gons, \(k \geq 7\). Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of \(k\)-gons, \(k \geq 7\), we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from \(k\)-barrels (also known as Löbel polytopes, whose graphs are biladders on \(2 k\) vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the \((5, 0)\)-nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons.
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 05C76
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 6916694
zbMATH Keywords: fullerenes
zbMATH Keywords: right-angled polytopes
zbMATH Keywords: truncation of edges
zbMATH Keywords: connected sum
zbMATH Keywords: \(k\)-belts
zbMATH Keywords: \(p\)-vector
zbMATH Keywords: cyclically 5-edge connected graph
describes a project that uses: House of Graphs
describes a project that uses: plantri
describes a project that uses: Fullerene
describes a project that uses: CaGe
describes a project that uses: FuiGui
MaRDI profile type: Publication
full work available at URL: sym10030067
OpenAlex ID: W2785856228
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Wikidata QID: Q130121920
DOI: 10.3390/SYM10030067
description: scientific article; zbMATH DE number 6916694