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Fullerenes and Belyi functions

Nikolai M. Adrianov, George B. Shabat

Abstract

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 the fullerene with the only hexagonal face is established by the methods of dessins d'enfants.

Fullerenes and Belyi functions

Abstract

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 the fullerene with the only hexagonal face is established by the methods of dessins d'enfants.
Paper Structure (9 sections, 2 theorems, 91 equations, 5 figures)

This paper contains 9 sections, 2 theorems, 91 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Fullerenes and dessins d'enfants
  3. Belyi functions corresponding to fullerenes
  4. Smallest fullerenes and their Belyi functions
  5. $p_6=0$. Dodecahedron
  6. $p_6=1$. Non-existing fullerene
  7. $p_6=2$. Barrel
  8. The geometry of one pentagonal face
  9. Discussion

Key Result

Theorem 1

There exist no dessins of genus $g=0$ with passport $( 3^k \,|\, 2^l \,|\, 5^m\,s^1 )$ for $s\ne 5$.

Figures (5)

  • Figure 1: Schlegel diagrams and the corresponding dessins.
  • Figure 2: Dodecahedron dessin and its factors by $\mathsf{C}_5$ and $\mathsf{D}_5$.
  • Figure 3: Two images of the dessin $D_{12}$.
  • Figure 4: Barrel dessin $D_{72}$ (a fulleren with two hexagons).
  • Figure 5: Flat approximation of face $A_1 A_7 A_{13} A_8 A_2$ of the barrel.

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 2