The fundamental group and Galois coverings of hexagonal systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-space

De La Peña, J. A.; Mendoza, L.

doi:https://doi.org/10.1155/IJMMS/2006/47381

International Journal of Mathematics and Mathematical Sciences

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 047381 | https://doi.org/10.1155/IJMMS/2006/47381

The fundamental group and Galois coverings of hexagonal systems in 3-space

J. A. De La Peña¹and L. Mendoza²

Received10 Aug 2005

Revised01 Aug 2006

Accepted11 Oct 2006

Published29 Nov 2006

Abstract

We consider hexagonal systems embedded into the 3-dimensional space ℝ3. We define the fundamental group π1(G) of such a system G and show that in case G is a finite hexagonal system with boundary, then π1(G) is a (non-Abelian) free group. In this case, the rank of π1(G) equals m(G)−h(G)−n(G)+1, where n(G) (resp., m(G), h(G)) denotes the number of vertices (resp., edges, hexagons) in G.

References

K. Bongartz and P. Gabriel, “Covering spaces in representation-theory,” Inventiones Mathematicae, vol. 65, no. 3, pp. 331–378, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application, vol. 87 of Pure and Applied Mathematics, Academic Press, New York, 1980.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. O. Dietrich-Buchecker and J.-P. Sauvage, “A synthetic molecular Trefoil knot,” Angewandte Chemie International Edition in English, vol. 28, no. 2, pp. 189–192, 1989.
View at: Publisher Site | Google Scholar
E. Flapan, When Topology Meets Chemistry. A Topological Look at Molecular Chirality, Outlooks, Cambridge University Press, Cambridge; Mathematical Association of America, Washington, DC, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Gabriel, “The universal cover of a representation-finite algebra,” in Representations of Algebras (Puebla, 1980), vol. 903 of Lecture Notes in Math., pp. 68–105, Springer, Berlin, 1981.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. J. F. Harris, Carbon Nanotubes and Related Structures: New Materials for the Twenty-First Century, Cambridge University Press, Cambridge, 1999.
View at: Google Scholar
R. Martínez-Villa and J. A. de la Peña, “The universal cover of a quiver with relations,” Journal of Pure and Applied Algebra, vol. 30, no. 3, pp. 277–292, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, New York, 1967.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Ch. Riedtmann, “Algebren, Darstellungsköcher, Überlagerungen und zurück,” Commentarii Mathematici Helvetici, vol. 55, no. 2, pp. 199–224, 1980.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. M. Walba, R. M. Richards, and R. Curtis Haltiwanger, “Total synthesis of the first molecular Möbius Band,” Journal of the American Chemical Society, vol. 104, no. 11, pp. 3219–3221, 1982.
View at: Publisher Site | Google Scholar
E. Wasserman, “Chemical topology,” Scientific American, vol. 207, no. 5, pp. 94–100, 1962.
View at: Google Scholar

Copyright

Copyright © 2006 Hindawi Publishing Corporation. 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.

PDF Download Citation Order printed copies

Views

173

Downloads

746

Citations