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Special Issue "Symmetry and Beauty of Knots"

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A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 December 2011)

Special Issue Editor

Guest Editor
Prof. Dr. Slavik V. Jablan

ICT College, 11000 Belgrade, Zdravka Celara 16, Serbia
Website | E-Mail
Fax: +381 11 2186105
Interests: theory of symmetry; antisymmetry; colored symmetry; mathematical crystallography; knot theory; math-art; ornamental art and design; modularity in science and art

Special Issue Information

Dear Colleagues,

Symmetry plays an important role in knot theory for example, every knot or a link has a symmetry group, which is much harder to determine than symmetries of solids since knots and links are considered up to ambient isotopy. Questions like establishing of general criteria for amphicheirality - existence of a left and right form of a knot or link, and invertibility - invariance of a knot under the change of its orientation are still open.

Knots and links appearing in nature have very high degree of summetry, therefore applications of knot theory in chemistry and biology are closely related to studying regular polyhedra (e.g., octahedron corresponding to the Borromean rings), geometry and topology of polyhedral DNA, or knotted Fullerenes - a fast developing area of research. Symmetrical knots and knot patterns such as Celtic knots, are the highlights in the history of art.

Contributions related to various aspects of connections between the theory of symmetry and knot theory are invited. Possible topics include, but are not limited to:

  • symmetry groups of knots, amphicheirality, invertbility and periodicity
  • symmetrical knots in chemistry, biology, art and architecture
  • knot patterns, friezes, Celtic knots, Sona sand drawings, Kolam patterns, decorative knots
  • symmetrical knots on different surfaces and virtual knots
  • knots and quantum computing
  • knots and polyhedra
  • knots and Fulerenes

Prof. Dr. Slavik Jablan
Guest Editor

Published Papers (8 papers)

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Research

Open AccessArticle Knots in Art
Symmetry 2012, 4(2), 302-328; doi:10.3390/sym4020302
Received: 9 May 2012 / Revised: 15 May 2012 / Accepted: 15 May 2012 / Published: 5 June 2012
Cited by 1
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
[...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Diagrammatics in Art and Mathematics
Symmetry 2012, 4(2), 285-301; doi:10.3390/sym4020285
Received: 8 March 2012 / Revised: 25 April 2012 / Accepted: 28 April 2012 / Published: 22 May 2012
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
[...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Figures

Open AccessArticle Following Knots down Their Energy Gradients
Symmetry 2012, 4(2), 276-284; doi:10.3390/sym4020276
Received: 3 February 2012 / Revised: 19 March 2012 / Accepted: 18 April 2012 / Published: 27 April 2012
Cited by 2 | HTML 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. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Classical Knot Theory
Symmetry 2012, 4(1), 225-250; doi:10.3390/sym4010225
Received: 3 February 2012 / Revised: 1 March 2012 / Accepted: 1 March 2012 / Published: 7 March 2012
Cited by 3
Abstract This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Intrinsic Symmetry Groups of Links with 8 and Fewer Crossings
Symmetry 2012, 4(1), 143-207; doi:10.3390/sym4010143
Received: 4 January 2012 / Revised: 18 January 2012 / Accepted: 31 January 2012 / Published: 20 February 2012
Cited by 4
Abstract
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but
[...] Read more.
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but one of these links and present explicit isotopies generating the symmetry group for every link. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle The 27 Possible Intrinsic Symmetry Groups of Two-Component Links
Symmetry 2012, 4(1), 129-142; doi:10.3390/sym4010129
Received: 13 January 2012 / Revised: 7 February 2012 / Accepted: 9 February 2012 / Published: 17 February 2012
PDF Full-text (313 KB)
Abstract
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(
[...] Read more.
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Figures

Open AccessArticle Knots on a Torus: A Model of the Elementary Particles
Symmetry 2012, 4(1), 39-115; doi:10.3390/sym4010039
Received: 16 November 2011 / Revised: 27 December 2011 / Accepted: 16 January 2012 / Published: 9 February 2012
PDF Full-text (7529 KB) | HTML Full-text | XML Full-text
Abstract
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin,
[...] Read more.
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori
Symmetry 2012, 4(1), 26-38; doi:10.3390/sym4010026
Received: 14 November 2011 / Revised: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012
Cited by 1 | PDF Full-text (192 KB)
Abstract
A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a
[...] Read more.
A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)

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