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Symmetry 2018, 10(3), 67; doi:10.3390/sym10030067

Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face

†
Steklov Mathematical Institute of Russian Academy of Sciences, 119991 Moscow, Russia
†
The author is a Young Russian Mathematics award winner.
Received: 19 January 2018 / Revised: 7 March 2018 / Accepted: 9 March 2018 / Published: 15 March 2018
(This article belongs to the Special Issue Mathematical Crystallography)
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Abstract

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 ≥ 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 ≥ 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. View Full-Text
Keywords: fullerenes; right-angled polytopes; truncation of edges; connected sum; k-belts; p-vector; cyclically 5-edge connected graph fullerenes; right-angled polytopes; truncation of edges; connected sum; k-belts; p-vector; cyclically 5-edge connected graph
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Erokhovets, N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. Symmetry 2018, 10, 67.

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