Fullerenes and Belyi functions

N. M. Adrianov ; G. B. Shabat

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 2, pp. 41-61

Résumé

The paper is an attempt to apply the theory of dessins d'enfants to the theory of fullerenes. The classical results concerning the calculation of the dodecahedron Belyi function are presented and then applied to the calculation of the Belyi function of the barrel, and the Euclidean geometry of the latter is investigated. The non-existence of a fullerene with the only hexagonal face is established by the methods of dessins d'enfants.

Bibliographie

[1] Adrianov N. M., “Gruppy monodromii i deistvie na $k$-podmnozhestvakh”, Fundament. i prikl. matem. (to appear)

[2] Bukhshtaber V. M., Erokhovets N. Yu., “Usecheniya prostykh mnogogrannikov i prilozheniya”, Tr. MIAN, 289 (2015), 115–144 | DOI | Zbl

[3] Bukhshtaber V. M., Erokhovets N. Yu., “Konstruktsii semeistv trekhmernykh mnogogrannikov, kharakteristicheskie fragmenty fullerenov i mnogogranniki Pogorelova”, Izv. RAN. Ser. matem., 81:5 (2017), 15–91 | DOI | MR | Zbl

[4] Velmin' V. P., “Rѣshenie neopredѣlennago uravneniya $u^m+v^n=w^k$”, Matem. sb., 24:4 (1904), 633–661

[5] Deza M., Dyutur Sikirich M., Shtogrin M. I., “Fullereny i disk-fullereny”, UMN, 68:4 (2013), 69–128 | DOI | MR | Zbl

[6] Ermakov' V. P., “Trekhchlennyya neopredѣlennyya uravneniya”, Matem. sb., 20:2 (1898), 293–298

[7] Zvonkin A. K., Lando S. K., Grafy na poverkhnostyakh i ikh prilozheniya, MTsNMO, M., 2010

[8] Klein F., Lektsii ob ikosaedre i reshenii uravnenii pyatoi stepeni, Nauka, M., 1989 | MR

[9] Mordukhai-Boltovskoi D. D., “Geometriya radiolyarii”, Uchenye zapiski Rostovskogo un-ta, 8, URSS, M., 1936, 1-91

[10] Arnaudies J.-M., Bertin J., Groupes, algèbres et géométrie, v. 3. Surfaces de Riemann, equation de Halphen et groupes polyédraux, Ellipses, Paris, 2001 | MR

[11] Benjamin S. C., Ardavan A., Briggs G. A. D., Britz D. A., Gunlycke D., Jefferson J., Jones M. A. G., Leigh D. F., Lovett B. W., Khlobystov A. N., Lyon S. A., Morton J. J. L., Porfyrakis K., Sambrook M. R., Tyryshkin A. M., “Towards a fullerene-based quantum computer”, J. Phys. Condens. Matter, 18:21 (2006), 867–883 | DOI

[12] Bowers P. L., Stephenson K., Uniformizing Dessins and Belyi Maps via Circle Packing, Mem. Amer. Math. Soc., 170, no. 805, Amer. Math. Soc., 2004 | MR

[13] Buchstaber V. M., Erokhovets N., Construction of fullerenes, 2015, arXiv: 1510.02948

[14] Chung F., Sternberg S., “Mathematics and the Buckyball”, American Scientist, 81:1 (1993), 56–71

[15] Couveignes J.-M., “Calcul et rationalité de fonctions de Belyi en genre 0”, Ann. Inst. Fourier, 44:1 (1994), 1–38 | DOI | MR | Zbl

[16] Girondo E., González-Diez G., Introduction to Compact Riemann Surfaces and Dessins d'Enfants, Cambridge Univ. Press, Cambridge, 2011 | MR

[17] Goldberg M., “The isoperimetric problem for polyhedra”, Tôhoku Math. J., 40 (1935), 226–236 | Zbl

[18] Grothendieck A., “Esquisse d'un Programme”, Geometric Galois Actions 1. Around Grothendieck's Esquisse d'un Programme, London Math. Soc. Lect. Note Ser., 242, eds. P. Lochak, L. Schneps, Cambridge Univ. Press, Cambridge, 1997, 5–48 | MR | Zbl

[19] Grünbaum B., “Graphs of polyhedra; polyhedra as graphs”, Discrete Math., 307 (2007), 445–463 | DOI | MR | Zbl

[20] Grünbaum B., Motzkin T. S., “The number of hexagons and the simplicity of geodesics on certain polyhedra”, Canad. J. Math., 15 (1963), 744–751 | DOI | MR | Zbl

[21] Halphen G. H., Sur la réduction des équations différentielles linéaires aux formes intégrables. Mémoires présentés par divers savants à l'Academie des sciences de l'Institut National de France (1883), v. III, Oeuvres de G.-H. Halphen, Paris, 1921, 1–260

[22] Izmestiev I., Color or cover, 2015, arXiv: 1503.00605

[23] Katz E., Geometrical analysis of radiolaria and fullerene structures: Who gets the credit?, Math. Intelligencer, 36:1 (2014), 34–36 | DOI | MR | Zbl

[24] Korkine A., “Sur l'impossibilité de la relation algébrique $x^n+y^n+z^n=0$”, C. R. Acad. Sci., 1880, 303–304 | Zbl

[25] Meija J., “Goldberg variations challenge”, Analytical Bioanalytical Chem., 385 (2006), 6–7 | DOI

[26] Meija J., “Solution to Goldberg variations challenge”, Analytical Bioanalytical Chem., 386 (2006), 4 pp.

[27] Petrovic D., Seke M., Srdjenovic B., Djordjevic A., “Applications of anti/prooxidant fullerenes in nanomedicine along with fullerenes influence on the immune system”, J. Nanomaterials, 2015, no. 2, 565638 | DOI

[28] Sijsling J., Voight J., “On computing Belyi maps”, Pub. Math. Besançon, 1 (2014), 73–131 | MR | Zbl

[29] Shabat G. B., Voevodsky V. A., “Drawing curves over number fields”, The Grothendieck Festschrift, v. III, Modern Birkhäuser Classics, 88, Birkhäuser, Boston, 2006, 199–227 | DOI | MR

[30] Steinitz E., “Polyeder und Raumeinteilungen”, Encycl. Math. Wiss., v. 3, Geometrie 3AB12, 1922, 1–139

[31] Schwarz H., “Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt”, J. Reine Angew. Math., 75 (1873), 292–335 | MR

[32] Vidunas R., Filipuk G., “A classification of coverings yielding Heun-to-hypergeometric reductions”, Osaka J. Math., 51:4 (2014), 867–905 | MR

[33] Thompson D'A. W., On Growth and Form, 2nd ed., Cambridge Univ. Press, 1917 | MR

[34] Voytekhovsky Y., Stepenshchikov D., “On the Motzkin–Grunbaum theorem”, Acta Cryst. A, 61:6 (2005), 584–585 | DOI | MR | Zbl