Special Issue "Symmetry and Beauty of Knots"

Guest Editor
Prof. Dr. Slavik V. Jablan
The Mathematical Institute, Knez Mihailova 35, P.O. Box 367, 1101 Belgrade, Serbia
Website: http://www.mi.sanu.ac.rs/~jablans/
E-Mail: sjablan@gmail.com
Phone: +381 11 2630170
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

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

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed Open Access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 300 CHF (Swiss Francs). English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.

Toru Ikeda Article: 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; in revised form: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012 Show/Hide Abstract | Download 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 regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ.

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Jack S. Avrin Article: Knots on a Torus: A Model of the Elementary Particles Symmetry 2012, 4(1), 39-115; doi:10.3390/sym4010039 Received: 16 November 2011; in revised form: 27 December 2011 / Accepted: 16 January 2012 / Published: 9 February 2012 Show/Hide Abstract | Download PDF Full-text (7529 KB) 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, 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.

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Jason Cantarella, James Cornish, Matt Mastin and Jason Parsley Article: The 27 Possible Intrinsic Symmetry Groups of Two-Component Links Symmetry 2012, 4(1), 129-142; doi:10.3390/sym4010129 Received: 13 January 2012; in revised form: 7 February 2012 / Accepted: 9 February 2012 / Published: 17 February 2012 Show/Hide Abstract | Download 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(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.

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Michael Berglund, Jason Cantarella, Meredith Perrie Casey, Eleanor Dannenberg, Whitney George, Aja Johnson, Amelia Kelley, Al LaPointe, Matt Mastin, Jason Parsley, Jacob Rooney and Rachel Whitaker Article: Intrinsic Symmetry Groups of Links with 8 and Fewer Crossings Symmetry 2012, 4(1), 143-207; doi:10.3390/sym4010143 Received: 4 January 2012; in revised form: 18 January 2012 / Accepted: 31 January 2012 / Published: 20 February 2012 Show/Hide Abstract | Download PDF Full-text (1600 KB) 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 one of these links and present explicit isotopies generating the symmetry group for every link.

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J. Scott Carter Article: Classical Knot Theory Symmetry 2012, 4(1), 225-250; doi:10.3390/sym4010225 Received: 3 February 2012; in revised form: 1 March 2012 / Accepted: 1 March 2012 / Published: 7 March 2012 Show/Hide Abstract | Download PDF Full-text (9080 KB) 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.

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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) | View HTML Full-text | Download PMC-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.

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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.

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Type of Paper: Article
Title: Knots on a Torus: a Model of the Elementary Particles 
Author: Jack S. Avrin
Affiliation: 28715 Leacrest DR. Rancho PalosVerdes, CA, USA; E-Mail: javrin@aol.com
Abstract: This paper summarizes a certain model of the elementary particles of physics. Although this model has been reported upon in previous publications, the unique emphasis here, in keeping with the nature of this journal, is on the fundamental symmetries that emerge in its development. Similarly, echoing the theme of this issue, we emphasize the model’s definitive relationship to knots; the “particles” of the model owe their very existence as topological solitons in spacetime, compatible with both Special and General Relativity to their ineradicable twist. Furthermore, each is endowed with the physical conformation of a generalized Moebius strip, otherwise expressible as a concatenation of a particular (2, n) torus knot or, alternatively, as the framing of an associated two-strand braid with closure. The model’s taxonomy and interactions then turn out to mirror, with minor deviation, those of the Standard Model but with increased simplification and explanatory power, due in good part to the fact that all particles in the model belong to a single topological genus. What is remarkable is how pervasive the influence of the initial choices of basic model parameters and conformation is upon not only the overall model that emerges but on its relationship to other areas of physics and mathematics.

Title of Paper: Symmetries of spatial graphs and rational twists along spheres and tori
Authors: Toru Ikeda *
Affiliation: Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi-Shi, Kochi 780-8520, Japan; E-Mail: ikedat@kochi-u.ac.jp
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 tubular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ.

Type of Paper: Article
Title: The 27 Possible Intrinsic Symmetry Groups of 2-Component Links
Authors: Jason Cantarella, James Cornish, Matt Mastin and Jason Parsley *
Affiliation: Department of Mathematics, Box 7588, Wake Forest University, Winston-Salem, NC 27109-7588, USA; Email: parslerj@wfu.edu
Abstract: We consider the 'intrinsic' symmetry group of a 2-component link L, defined to be the image σ(L) of the natural homomorphism from the standard symmetry group MCG(S³,L) to the product MCG(S³) x MCG(L). The group σ(L), 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 isotopies. For of 2-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.

Title: Intrinsic Symmetry Groups of Links with 8 and fewer crossings
Authors: Michael Berglund 1, Jason Cantarella 2, Meredith Perrie Casey 1, Eleanor Dannenberg 1, Whitney George 1, Aja Johnson 1, Amelia Kelley 1, Al LaPointe 1, Matt Mastin 1, Jason Parsley 3*, Jacob Rooney 1, Rachel Whitaker 1
Affiliations:
1 University of Georgia, Mathematics Department (student), Boyd GSRC, Athens, GA, USA
2 University of Georgia, Mathematics Department, Boyd GSRC, Athens, GA USA
3 Wake Forest University, Mathematics Department, 127 Manchester Hall, Winston-Salem, NC, USA; E-mail: rjasonparsley@gmail.com
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 one of these links and present explicit isotopies generating the symmetry group for every link.
Keywords: knot; symmetry group of knot; link symmetry; Whitten group