# Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face

^{†}

## Abstract

**:**

## 1. Introduction

**Proposition**

**1.**

**Theorem 1**(Steinitz)

**.**

**Theorem**

**2.**

**Example**

**1.**

**Theorem 4**(The first main result)

**.**

**Remark**

**1.**

**Theorem**

**5**

**.**A simple 3-polytope P is a $Pog$-polytope if and only if either P is a q-barrel, $q\ge 5$, or it can be constructed from the 5- or the 6-barrel by a sequence of $(2,k)$-truncations (Figure 8a), $k\ge 6$, and connected sums with the 5-barrel (Figure 8b).

**Theorem**

**6**

**.**Any fullerene $P\in \mathcal{F}\backslash {\mathcal{F}}^{\ast}=\mathcal{D}\backslash \{{D}_{0}\}$ can be constructed from the 5-barrel by operations of a connected sum with a copy of the 5-barrel along the center of a patch ${C}_{1}$. It cannot be obtained from a simple 3-polytope without 4-gons by a $(2,k)$-truncation, $k\ge 6$.

**Proposition**

**2.**

**Theorem 7**(The second main result)

**.**

**Theorem 8**(The third main result)

**.**

## 2. Proof of the Main Results

**Proof of the first main result**

**(Theorem 4).**

**Corollary**

**1.**

**Proof.**

**Example**

**2.**

**Remark**

**2.**

**Theorem**

**9.**

**Proof of the second main result**

**(Theorem 7).**

**Proof**

**of Proposition 2.**

**Corollary**

**2.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Proof of the third main result**

**(Theorem 8).**

**Lemma**

**8.**

**Proof.**

## 3. Discussion

## 4. Prospects

- The result of Theorem 8 may be strengthened. It seems that the operation of a $(2,7;6,6)$-truncation can be eliminated. Furthermore, it seems to be an open question whether there is a finite set of growth operations transforming the family ${\mathcal{P}}_{\le 7}$ to itself sufficient to reduce any polytope in ${\mathcal{P}}_{7}$ with all the non-hexagons isolated to some polytope in ${\mathcal{P}}_{\le 7}$. Let us remind that due to results in [51], there are no finite sets of growth operations transforming fullerenes to fullerenes sufficient to reduce any fullerene with all 5-gons isolated to some fullerene.
- There arise further questions about p-vectors of $Pog$-polytopes. For example, for given numbers $({p}_{k},k\ge 7)$ for which values of ${p}_{6}$ does a $Pog$-polytope realizing this p-vector exist?
- To apply the construction of fullerenes and $Pog$-polytopes by operations presented in this article to problems in polytope theory, toric topology and hyperbolic geometry; for example, to give a new proof of the four color theorem for special classes of $Pog$-polytopes; or for a given $Pog$-polytope to enumerate all characteristic mappings $\Lambda $ and ${\Lambda}_{2}$. There is a question about describing the transformation of differential-geometric and algebraic-topological properties of the manifolds under transformation of polytopes.
- To estimate the numbers of polytopes in ${\mathcal{P}}_{7}$ and ${\mathcal{P}}_{7,5}$ with the given number of faces.

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\mathcal{F}$ | the family of fullerenes |

${\mathcal{P}}_{7}$ | the family of simple 3-polytopes with 5-, 6- and one 7-gonal face |

${\mathcal{P}}_{7,5}$ | the subfamily in ${\mathcal{P}}_{7}$ consisting of polytopes with the 7-gon adjacent to a 5-gon |

${\mathcal{P}}_{\le 7,5}$ | $\mathcal{F}\bigsqcup {\mathcal{P}}_{7,5}$ |

${\mathcal{P}}_{\le 7}$ | $\mathcal{F}\bigsqcup {\mathcal{P}}_{7}$ |

$\mathcal{D}$ | the family of polytopes consisting of the dodecahedron and the $(5,0)$-nanotubes |

$Pog$-polytope | Pogorelov polytope |

$Po{g}^{\ast}$-polytope | Pogorelov polytope with any 5-belt surrounding a face |

$ck$-connected | cyclically k-edge connected |

${c}^{\ast}k$-connected | strongly cyclically k-edge connected |

${\mathcal{A}}^{\ast}$ | the subfamily of all $Po{g}^{\ast}$-polytopes in a family $\mathcal{A}$ |

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**Figure 11.**A polytope in ${\mathcal{P}}_{7,5}$, which cannot be obtained from a polytope in ${\mathcal{P}}_{\le 7,5}$ by a $(2,k)$-truncation or a connected sum with the 5-barrel.

**Figure 14.**(

**a**) An initial disk; (

**b**) the addition of belts; (

**c**) a construction of the complementary disk.

**Figure 15.**The first disk for the case ${p}_{7}=2$, ${p}_{k}=0$, $k\ge 8$. The second disk is drawn in Figure 14c.

**Figure 16.**(

**a**) A patch ${C}_{1}$; (

**b**) an operation inverse to a connected sum; (

**c**) a non-existing patch.

**Figure 19.**(

**a**) The 7-gon adjacent to two subsequent 5-gons; (

**b**) the patch ${D}_{2}$; (

**c**) the patch ${D}_{3}$.

**Figure 20.**(

**a**) The 7-gon adjacent to a 5-gon; (

**b**) the patch ${D}_{1}$; (

**c**) the case when ${F}_{b}$ is a 5-gon; (

**d**) the case when ${F}_{a}$ and ${F}_{c}$ are 5-gons.

**Figure 21.**(

**a**) The case when ${F}_{a}$ is a 5-gon and ${F}_{c}$ is a 6-gon; (

**b**) the patch ${D}_{1}$; (

**c**) the case when ${F}_{d}$ is a 5-gon.

**Figure 22.**(

**a**) The case when ${F}_{u}$ is a 5-gon; (

**b**) the case when ${F}_{u}$ is a 6-gon; (

**c**) the patch ${D}_{1}$.

**Figure 23.**(

**a**) Four 5-gons; (

**b**) ${F}_{p}$ and ${F}_{v}$ are 6-gons; (

**c**) ${F}_{p}$ and ${F}_{u}$ are 6-gons.

**Figure 24.**(

**a**) ${F}_{p}$ and ${F}_{v}$ are 6-gons; (

**b**) the patch ${D}_{2}$; (

**c**) the patch ${D}_{3}$.

**Figure 25.**(

**a**) ${F}_{p}$ and ${F}_{u}$ are 6-gons; (

**b**) ${F}_{w}$ is a 5-gon; (

**c**) ${F}_{t}$ is a 5-gon; (

**d**) ${F}_{w}$ is a 6-gon.

**Figure 27.**(

**a**) The beginning of the patch D; (

**b**) a transformation of a neighborhood of ${C}_{1}$; (

**c**) ${F}_{s}$ is a 5-gon; (

**d**) the straightening along the edge ${F}_{p}\cap {F}_{s}$.

**Figure 28.**(

**a**) ${F}_{s}$ is a 6-gon; (

**b**) ${F}_{r}$ and ${F}_{j}$ are 6-gons; (

**c**) the reduction of the patch ${D}_{2}$.

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**MDPI and ACS Style**

Erokhovets, N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. *Symmetry* **2018**, *10*, 67.
https://doi.org/10.3390/sym10030067

**AMA Style**

Erokhovets N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. *Symmetry*. 2018; 10(3):67.
https://doi.org/10.3390/sym10030067

**Chicago/Turabian Style**

Erokhovets, Nikolai. 2018. "Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face" *Symmetry* 10, no. 3: 67.
https://doi.org/10.3390/sym10030067