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  • 1 The Mathematical Institute Knez Mihailova 36 P.O. Box 367 11001 Belgrade Serbia
  • | 2 Faculty of Mechanical Engineering A. Medvedeva 14 18000 Niš Serbia
  • | 3 University of Pennsylvania 209 South 33rd Street Philadelphia PA 19104-6395 USA
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This paper contains a survey of different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture chemistry, biology (or even jewelry). We describe several ways of obtaining 4-valent polyhedral graphs and their corresponding knots and links from basic polyhedra: mid-edge construction, cross-curve and double-line covering, and edge doubling construction. These methods are implemented in the Mathematica-based program LinKnot and can be applied to the data base of basic polyhedra. In a similar way, an edge doubling construction transforms fullerene graphs into alternating knot and link diagrams. In the last part of the paper is proposed the use of virtual knots and links and the corresponding nonplanar graphs obtained from their Gauss codes.

  • Sauvage J.P., Dietrich-Buckecker C. Molecular catenanes, rotaxanes and knots, New York, Wiley, 1999.

    Dietrich-Buckecker C. , '', in Molecular catenanes, rotaxanes and knots , (1999 ) -.

  • Flapan E. When topology meets chemistry: A look at molecular chemistry, Cambridge University Press, Cambridge, 2000.

    Flapan E. , '', in When topology meets chemistry: A look at molecular chemistry , (2000 ) -.

  • Vallisser T. Other geometries in architecture: bubbles, knots and minimal surfaces http://www.springer.com/cda/content/document/cda_downloaddocument/978-88-470-1121-2_Wallisser.pdf

  • Hoste J., Thistlethwaite M., Knotscape, http://www.math.utk.edu/~morwen/knotscape.html (M.Thistlethwaite, Private communication, 10. Feb. 2010)

  • Zhang Y., Seeman N.C., Construction of a DNA-truncated octahedron, J. Am. Chem. Soc. Vol. 116, 1994, pp. 1661–1669.

    Seeman N.C. , 'Construction of a DNA-truncated octahedron ' (1994 ) 116 J. Am. Chem. Soc. : 1661 -1669.

    • Search Google Scholar
  • Qiu W. Y., Zhai X. D. Molecular design of Goldberg polyhedral links, J. Mol. Struc. (Theochem) Vol. 756, 2005, pp. 163–166.

    Zhai X. D. , 'Molecular design of Goldberg polyhedral links ' (2005 ) 756 J. Mol. Struc. (Theochem) : 163 -166.

    • Search Google Scholar
  • Qiu W. Y., Zhai X. D., Qiu Y. Y. Architecture of Platonic and Archimedean polyhedral links, Science in China, Ser. B: Chemistry, Vol. 51, No. 1, 2008, pp. 13–18.

    Qiu Y. Y. , 'Architecture of Platonic and Archimedean polyhedral links ' (2008 ) 51 Science in China, Ser. B: Chemistry : 13 -18.

    • Search Google Scholar
  • Conway J. An enumeration of knots and links and some of their related properties, in Computational Problems in Abstract Algebra, ed. J. Leech, Proc. Conf. Oxford 1967, (Pergamon Press, New York, 1970), pp. 329–358.

    Conway J. , '', in Computational Problems in Abstract Algebra , (1970 ) -.

  • Caudron A. Classification des noeuds et des enlancements, Public. Math. d’Orsay 82. Univ. Paris Sud, Dept. Math., Orsay, 1982.

    Caudron A. , '', in Classification des noeuds et des enlancements , (1982 ) -.

  • Jablan S. V., Sazdanović R LinKnot — Knot Theory by Computer, World Scientific, New Jersey, London, Singapore, 2007; http://math.ict.edu.rs/ , http://www.mi.sanu.ac.rs/vismath/linknot/index.html

    Sazdanović R. , '', in LinKnot — Knot Theory by Computer , (2007 ) -.

  • Jablan S., Radović Lj., Sazdanović R. Pyramidal knots and links and their invariants, MATCH Commun. Math. Comput. Chem., Vol. 65, No. 3, 2011, pp. 541–580.

    Sazdanović R. , 'Pyramidal knots and links and their invariants ' (2011 ) 65 MATCH Commun. Math. Comput. Chem. : 541 -580.

    • Search Google Scholar
  • Jablan S., Radović Lj., Sazdanović R. Knots and links derived from prismatic graphs, MATCH Commun. Math. Comput. Chem,, Vol. 66, No. 1, 2011, pp. 65–92.

    Sazdanović R. , 'Knots and links derived from prismatic graphs ' (2011 ) 66 MATCH Commun. Math. Comput. Chem : 65 -92.

    • Search Google Scholar
  • Fowler P. D., Manolopoulos D.E. An atlas of fullerenes, Oxford University Press, Oxford, 1995.

    Manolopoulos D.E. , '', in An atlas of fullerenes , (1995 ) -.

  • van Wijk J. J., Seifert View, http://www.win.tue.nl/~vanwijk/seifertview/

  • Kauffman L. H., Virtual knots, Talks at MSRI Meeting, January 1997 and AMS Meeting, at University of Maryland, College Park, March 1997.

  • Kauffman L. H., Virtual Knot Theory, Europ. J. Combinatorics, Vol. 20, 1999, pp. 663–691.

    Kauffman L. H. , 'Virtual Knot Theory ' (1999 ) 20 Europ. J. Combinatorics : 663 -691.

  • Kozlov D. Topological method of construction of point surfaces as physical models, http://www.marhi.ru/AMIT/2008/spec08/papers/Kozlov/Kozlov02_paper_EAEA2007.pdf

  • Grossman B., Bathsheba Sculpture, http://www.bathsheba.com/

  • Burt M. Periodical sponge structures and uniform sponge polyhedra in nature and in the realm of the theoretically imaginable, Vismath, Vol. 9, No. 4, 2007. http://www.mi.sanu.ac.rs/vismath/burt/index.html

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