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

doi:10.3390/sym10030067

Choose your preferred view mode

Please select whether you prefer to view the MDPI pages with a view tailored for mobile displays or to view the MDPI pages in the normal scrollable desktop version. This selection will be stored into your cookies and used automatically in next visits. You can also change the view style at any point from the main header when using the pages with your mobile device.

Use mobile version Use desktop version

You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.

MDPI — Symmetry

Symmetry

MDPI
Journals A-Z
Information & Guidelines
Initiatives
- Sciforum
- Preprints
- Scilit
- MDPI Books
- Encyclopedia
About
Login
Register
Submit
Switch to Desktop Version
Close

MDPI
Journals A-Z
Information & Guidelines
Initiatives
- - Sciforum
  - Preprints
  - Scilit
  - MDPI Books
  - Encyclopedia
About

Login
Register
Submit

Title / Keyword

Author / Affiliation

Journal

Article Type

Advanced

IMPACT
FACTOR
1.256

Volume 10, Issue 3

► ▼ Article Menu

Views

Downloads

No citations found yet 0

Metrics 0

Article Versions

Abstract
Full-Text PDF [636 KB]
Full-Text HTML
Full-Text XML
Full-Text Epub
Article Versions Notes

Related Info

Google Scholar
Order Reprints

More by Authors

on DOAJ

Erokhovets, N

on Google Scholar

Erokhovets, N

on PubMed

Erokhovets, N

Export Article

BibTeX
EndNote
RIS

Create a SciFeed alert for new publications

With following keywords

fullerenes

right-angled polytopes

truncation of edges

connected sum

k-belts

p-vector

cyclically 5-edge connected graph

By following author

Nikolai Erokhovets

Advanced options

Email:

Freq:

One email with all search results

One email for each search

Renew

/ajax/scifeed/subscribe

MOL Viewer

Open AccessFeature PaperArticle

Symmetry 2018, 10(3), 67; https://doi.org/10.3390/sym10030067

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

Nikolai Erokhovets^†

Steklov Mathematical Institute of Russian Academy of Sciences, 119991 Moscow, Russia

^†

The author is a Young Russian Mathematics award winner.

Received: 19 January 2018 / Revised: 7 March 2018 / Accepted: 9 March 2018 / Published: 15 March 2018

(This article belongs to the Special Issue Mathematical Crystallography)

View Full-Text | Download PDF [636 KB, uploaded 15 March 2018] |

Browse Figures

Abstract

A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed numbers of k-gons,

k \geq 7

. Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of k-gons,

k \geq 7

, we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from k-barrels (also known as Löbel polytopes, whose graphs are biladders on

2 k

vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the

(5, 0)

-nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons. View Full-Text

Keywords: fullerenes; right-angled polytopes; truncation of edges; connected sum; k-belts; p-vector; cyclically 5-edge connected graph fullerenes; right-angled polytopes; truncation of edges; connected sum; k-belts; p-vector; cyclically 5-edge connected graph

►▼ Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Share & Cite This Article

MDPI and ACS Style

Erokhovets, N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. Symmetry 2018, 10, 67.

AMA Style

Erokhovets N. Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. Symmetry. 2018; 10(3):67.

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.

Find Other Styles

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Metrics

For more information on the journal statistics, click here. Multiple requests from the same IP address are counted as one view.

Article Access Statistics

Abstract views Pdf views Html views

Comments

[Return to top]

Submit to Symmetry Review for Symmetry Edit a Special Issue

Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert

Subscribe to receive issue release notifications and newsletters from MDPI journals

Select Journal/Journals:

Volume 10, Issue 3

Article Versions

Related Info

More by Authors

Export Article

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

Abstract

Share & Cite This Article

Related Articles

Article Metrics

Article Access Statistics

Comments