Fullerenes, Polytopes and Toric Topology

Buchstaber, Victor Matveevich; Erokhovets, Nickolai

doi:10.1142/9789813226579_0002

Cited by 21 publications

(21 citation statements)

References 30 publications

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“…A polytope from the class P has neither triangular nor quadrangular facets. The Pogorelov class contains all fullerenes; this follows from the results of Došlić [32] (see also [8,Corollary 3.16] and [9,10]). As we mentioned in the Introduction, the results of [60] imply that the number of combinatorially different fullerenes with p 6 hexagonal facets grows as p 9 6 .…”

Section: Small Coversmentioning

confidence: 63%

“…Theorem B.1 (see [10]). A simple graph on a 2-dimensional sphere is the graph of a convex 3-polytope if and only if the following two conditions are satisfied:…”

Section: Appendix B Combinatorics and Constructions Of Pogorelov Polmentioning

confidence: 99%

“…As we mentioned in the Introduction, the results of [60] imply that the number of combinatorially different fullerenes with p 6 hexagonal facets grows as p 9 6 . We also note that the class P contains simple 3-polytopes with pentagonal, hexagonal and one heptagonal facet, which are used in the construction of fullerenes by means of truncations (see [9,10,11]). Finally, we show in Corollary B.14 that for any finite sequence of nonnegative integers p k , k 7, there exists a Pogorelov polytope whose number of k-gonal facets is p k .…”

Section: Small Coversmentioning

confidence: 99%

“…After reducing the statement to analysing the cohomology of moment-angle manifolds, we apply several nontrivial combinatorial and algebraic lemmata of Fan, Ma and Wang [33,34], used in their proof of cohomological rigidity for moment-angle compexes of flag 2-spheres without chordless cycles of length 4. Families of polytopes from the class P also feature in the works [9,10] on combinatorial constructions of fullerenes.…”

Section: Introductionmentioning

confidence: 97%

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Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Buchstaber¹,

Erokhovets²,

Masuda³

et al. 2017

Russ. Math. Surv.

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Abstract. A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes.We consider the class P of 3-dimensional combinatorial simple polytopes P , different from a tetrahedron, whose facets do not form 3-and 4-belts. This class includes mathematical fullerenes, i. e. simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from P admits a right-angled realisation in Lobachevsky 3-space, which is unique up to isometry.Our families of smooth manifolds are associated with polytopes from the class P. The first family consists of 3-dimensional small covers of polytopes from P, or hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes from P. Our main result is that both families are cohomologically rigid, i. e. two manifolds M and M from either of the families are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that if M and M are diffeomorphic, then their corresponding polytopes P and P are combinatorially equivalent. These results are intertwined with the classical subjects of geometry and topology, such as combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.

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Section: Small Coversmentioning

confidence: 63%

“…Theorem B.1 (see [10]). A simple graph on a 2-dimensional sphere is the graph of a convex 3-polytope if and only if the following two conditions are satisfied:…”

Section: Appendix B Combinatorics and Constructions Of Pogorelov Polmentioning

confidence: 99%

Section: Small Coversmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Buchstaber¹,

Erokhovets²,

Masuda³

et al. 2017

Russ. Math. Surv.

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“…If a fullerene P has no adjacent pentagons, then the proof of Theorem 9.12 in [28] implies that P is obtained from a polytope in F 1 by operation A 6 , which is a (2; 7; 5, 5)truncation.…”

Section: Lemmamentioning

confidence: 99%

Finite sets of operations sufficient to construct any fullerene from C 20

Buchstaber

Erokhovets

2016

Struct Chem

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We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene C 20 ). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge-and two edges-truncations.

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Spectral clustering of combinatorial fullerene isomers based on their facet graph structure

2020

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After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene’s stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were performed on all isomers of C40, C60 and C80. This paper suggests a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We also show that our classification scheme can serve as a formal stability criterion, which became evident from a comparison of our results with recent quantum chemical calculations (Sure et al. in Phys Chem Chem Phys 19:14296–14305, 2017). We apply our method to classify all isomers of C60 and give an example of two different cospectral isomers of C44. Calculations are done with our own Python scripts available at (Bille et al. in Fullerene database and classification software, https://www.uni-ulm.de/mawi/mawi-stochastik/forschung/fullerene-database/, 2020). The only input for our algorithm is the vector of positions of pentagons in the facet spiral. These vectors and Schlegel diagrams are generated with the software package Fullerene (Schwerdtfeger et al. in J Comput Chem 34:1508–1526, 2013).

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Fullerenes, Polytopes and Toric Topology

Cited by 21 publications

References 30 publications

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Finite sets of operations sufficient to construct any fullerene from C 20

Spectral clustering of combinatorial fullerene isomers based on their facet graph structure

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