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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 5, Pages 15–91 (Mi izv8665)  

This article is cited in 7 scientific papers (total in 7 papers)

Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes

V. M. Buchstaber, N. Yu. Erokhovets

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We describe the combinatorics of three families of simple 3-dimensional polytopes which play an important role in various problems of algebraic topology, hyperbolic geometry, graph theory, and their applications. The first family $\mathcal{P}_{\leqslant 6}$ consists of simple polytopes with at most hexagonal faces. The second family $\mathcal{P}_\mathrm{pog}$ consists of Pogorelov polytopes. The third family $\mathcal{F}$ consists of fullerenes and is the intersection of the first two. We show that in the case of fullerenes there are stronger results than for the first two. Our main tools are $k$-belts of faces, simple partitions of a disc and the operations of transformation and connected sum.

Keywords: fullerene, nanotube, Pogorelov polytope, partition of a disc, operations of cutting off edges, operations of connected sum and addition of a belt, patches, $k$-belts.

Funding Agency Grant Number
Russian Science Foundation 14-11-00414
This work is supported by the Russian Science Foundation under grant no. 14-11-00414.


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English version:
Izvestiya: Mathematics, 2017, 81:5, 901–972

Bibliographic databases:

UDC: 514.172.45
MSC: 05C75, 52B10, 92E10
Received: 14.02.2017
Revised: 15.04.2017

Citation: V. M. Buchstaber, N. Yu. Erokhovets, “Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes”, Izv. RAN. Ser. Mat., 81:5 (2017), 15–91; Izv. Math., 81:5 (2017), 901–972

Citation in format AMSBIB
\by V.~M.~Buchstaber, N.~Yu.~Erokhovets
\paper Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes
\jour Izv. RAN. Ser. Mat.
\yr 2017
\vol 81
\issue 5
\pages 15--91
\jour Izv. Math.
\yr 2017
\vol 81
\issue 5
\pages 901--972

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    This publication is cited in the following articles:
    1. V. M. Buchstaber, N. Yu. Erokhovets, “Fullerenes, polytopes and toric topology”, Combinatorial and toric homotopy, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 35, World Sci. Publ., Hackensack, NJ, 2018, 67–178  crossref  mathscinet  zmath  isi  scopus
    2. N. Erokhovets, “Construction of fullerenes and Pogorelov polytopes with 5-, 6- and one 7-gonal face”, Symmetry, 10:3 (2018), 67, 28 pp.  crossref  zmath  isi  scopus
    3. B. Hua, Y. Su, “The set of vertices with positive curvature in a planar graph with nonnegative curvature”, Adv. Math., 343 (2019), 789–820  crossref  mathscinet  zmath  isi  scopus
    4. X. Yu, “Optimized development model of rural revitalization based on sustainable agricultural development”, Rev. Fac. Agron., 36:4 (2019), 1069–1077  isi
    5. V. M. Buchstaber, I. Yu. Limonchenko, “Massey products, toric topology and combinatorics of polytopes”, Izv. Math., 83:6 (2019), 1081–1136  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. N. Yu. Erokhovets, “Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions”, Proc. Steklov Inst. Math., 305 (2019), 78–134  mathnet  crossref  crossref  isi  elib
    7. G. Brinkmann, S. Van den Eynde, “Patches with short boundaries”, European J. Combin., 81 (2019), 285–297  crossref  mathscinet  zmath  isi  scopus
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