Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 19 January 2018 / Approved: 31 January 2018 / Online: 31 January 2018 (02:06:27 CET)

A peer-reviewed article of this Preprint also exists.

Erokhovets, N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. Symmetry 2018, 10, 67. Erokhovets, N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. Symmetry 2018, 10, 67.

Journal reference: Symmetry 2018, 10, 67
DOI: 10.3390/sym10030067

Abstract

A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. It has no 3- and 4-gons and may have any prescribed numbers of k-gons, k ≥ 7. Any 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 polytopes. Any Pogorelov polytope different from Löbel polytopes can be constructed from the 5- or the 6-barrel by cuttings 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 cuttings 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 can not 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 cuttings off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons.

Subject Areas

fullerenes; right-angled polytopes; truncation of edges; connected sum; k-belts; p-vector

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