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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 4(412), Pages 69–128 (Mi umn9544)  

This article is cited in 10 scientific papers (total in 10 papers)

Fullerenes and disk-fullerenes

M. Dezaa, M. Dutour Sikirićb, M. I. Shtogrincd

a École Normale Supérieure, Paris, France
b Rudjer Bošković Institute, Zagreb, Croatia
c Demidov Yaroslavl State University, Yaroslavl, Russia
d Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: A geometric fullerene, or simply a fullerene, is the surface of a simple closed convex 3-dimensional polyhedron with only 5- and 6-gonal faces. Fullerenes are geometric models for chemical fullerenes, which form an important class of organic molecules. These molecules have been studied intensively in chemistry, physics, crystallography, and so on, and their study has led to the appearance of a vast literature on fullerenes in mathematical chemistry and combinatorial and applied geometry. In particular, several generalizations of the notion of a fullerene have been given, aiming at various applications. Here a new generalization of this notion is proposed: an $n$-disk-fullerene. It is obtained from the surface of a closed convex 3-dimensional polyhedron which has one $n$-gonal face and all other faces 5- and 6-gonal, by removing the $n$-gonal face. Only 5- and 6-disk-fullerenes correspond to geometric fullerenes. The notion of a geometric fullerene is therefore generalized from spheres to compact simply connected two-dimensional manifolds with boundary. A two-dimensional surface is said to be unshrinkable if it does not contain belts, that is, simple cycles consisting of 6-gons each of which has two neighbours adjacent at a pair of opposite edges. Shrinkability of fullerenes and $n$-disk-fullerenes is investigated.
Bibliography: 87 titles.

Keywords: polygon, convex polyhedron, planar graph, fullerene, patch, disk-fullerene.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 11.G34.31.0053
НШ-4995.2012.1
Russian Foundation for Basic Research 11-01-00633
Alexander von Humboldt-Stiftung


DOI: https://doi.org/10.4213/rm9544

Full text: PDF file (1052 kB)
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English version:
Russian Mathematical Surveys, 2013, 68:4, 665–720

Bibliographic databases:

UDC: 514.172.45+515.164+519.17
MSC: 52A15, 57M20, 05C10
Received: 11.10.2012

Citation: M. Deza, M. Dutour Sikirić, M. I. Shtogrin, “Fullerenes and disk-fullerenes”, Uspekhi Mat. Nauk, 68:4(412) (2013), 69–128; Russian Math. Surveys, 68:4 (2013), 665–720

Citation in format AMSBIB
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    Erratum

    This publication is cited in the following articles:
    1. Yang Gao, Heping Zhang, “Clar structure and Fries set of fullerenes and (4,6)-fullerenes on surfaces”, J. Appl. Math., 2014 (2014), 196792, 11 pp.  crossref  mathscinet  isi  scopus
    2. T. Yu. Nikolaienko, E. S. Kryachko, “Formation of dimers of light noble atoms under encapsulation within fullerene's voids”, Nanoscale Res. Lett., 10 (2015), 185, 9 pp.  crossref  isi  elib  scopus
    3. V. M. Buchstaber, N. Yu. Erokhovets, “Truncations of simple polytopes and applications”, Proc. Steklov Inst. Math., 289 (2015), 104–133  mathnet  crossref  crossref  isi  elib
    4. N. Yu. Erokhovets, “$k$-poyasa i rebernye tsikly trekhmernykh prostykh mnogogrannikov s ne bolee chem shestiugolnymi granyami”, Dalnevost. matem. zhurn., 15:2 (2015), 197–213  mathnet  elib
    5. N. V. Prudnikova, “Konstruktsii fullerenov s chislom shestiugolnikov ne bolshe 7”, Dalnevost. matem. zhurn., 15:2 (2015), 247–263  mathnet  elib
    6. V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. V. M. Buchstaber, N. Yu. Erokhovets, “Finite sets of operations sufficient to construct any fullerene from $C_{20}$”, Struct. Chem., 28:1 (2017), 225–234  mathnet  crossref  mathscinet  isi  scopus
    8. V. M. Buchstaber, N. Yu. Erokhovets, “Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes”, Izv. Math., 81:5 (2017), 901–972  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. V. M. Buchstaber, N. Yu. Erokhovets, “Fullerenes, polytopes and toric topology”, Combinatorial and Toric Homotopy: Introductory Lectures, Lecture Notes Series Institute For Mathematical Sciences National University of Singapore, 35, ed. A. Darby, J. Grbic, Z. Lu, J. Wu, World Scientific Publ Co Pte Ltd, 2018, 67–178  crossref  isi
    10. N. Erokhovets, “Construction of fullerenes and Pogorelov polytopes with 5-, 6-and one 7-gonal face”, Symmetry-Basel, 10:3 (2018), 67, 28 pp.  crossref  isi
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