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Table of Contents

Symmetry, Volume 4, Issue 2 (June 2012), Pages 265-335

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	p. 265-275 Steve Wilson Article: Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators Symmetry 2012, 4(2), 265-275; doi:10.3390/sym4020265 Received: 29 January 2012; in revised form: 26 March 2012 / Accepted: 5 April 2012 / Published: 16 April 2012 Show/Hide Abstract | Download PDF Full-text (262 KB) Abstract: This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group. (This article belongs to the Special Issue Polyhedra)
	p. 276-284 Louis H. Kauffman Article: Following Knots down Their Energy Gradients Symmetry 2012, 4(2), 276-284; doi:10.3390/sym4020276 Received: 3 February 2012; in revised form: 19 March 2012 / Accepted: 18 April 2012 / Published: 27 April 2012 Show/Hide Abstract | Download PDF Full-text (325 KB) | Download XML Full-text Abstract: This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. (This article belongs to the Special Issue Symmetry and Beauty of Knots)
	p. 285-301 Radmila Sazdanovic Article: Diagrammatics in Art and Mathematics Symmetry 2012, 4(2), 285-301; doi:10.3390/sym4020285 Received: 8 March 2012; in revised form: 25 April 2012 / Accepted: 28 April 2012 / Published: 22 May 2012 Show/Hide Abstract | Download PDF Full-text (30730 KB) Abstract: This paper explores two-way relations between visualizations in mathematics and mathematical art, as well as art in general. A collection of vignettes illustrates connection points, including visualizing higher dimensions, tessellations, knots and links, plotting zeros of polynomials, and new and rapidly developing mathematical discipline, diagrammatic categorification. (This article belongs to the Special Issue Symmetry and Beauty of Knots)
	p. 302-328 Slavik Jablan, Ljiljana Radović, Radmila Sazdanović and Ana Zeković Article: Knots in Art Symmetry 2012, 4(2), 302-328; doi:10.3390/sym4020302 Received: 9 May 2012; in revised form: 15 May 2012 / Accepted: 15 May 2012 / Published: 5 June 2012 Show/Hide Abstract | Download PDF Full-text (5009 KB) Abstract: We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture. (This article belongs to the Special Issue Symmetry and Beauty of Knots)
	p. 329-335 Allen D. Parks Article: Topological Invariance under Line Graph Transformations Symmetry 2012, 4(2), 329-335; doi:10.3390/sym4020329 Received: 6 April 2012; in revised form: 1 June 2012 / Accepted: 4 June 2012 / Published: 8 June 2012 Show/Hide Abstract | Download PDF Full-text (177 KB) | Download XML Full-text Abstract: It is shown that the line graph transformation G ↦ L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of L(G). As a consequence, the homology of K is isomorphic to that of G. This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of G as a line graph transformation invariant.
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