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Symmetry
Special Issues
Knot Theory and Its Applications
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Knot Theory and Its Applications

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

Deadline for manuscript submissions: closed (31 August 2017) | Viewed by 22103

Special Issue Editor

Prof. Dr. Louis H. Kauffman

E-Mail Website
Guest Editor
Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago, Chicago, IL 60607-7045, USA
Interests: geometric topology; classical knot theory; virtual knot theory; higher dimensional knot theory; quantum knots; topological quantum field theory; quantum computing; topological quantum computing; diagrammatic and categorical approaches to mathematical structure
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleague,

You are invited to contribute a paper to a Special Issue of Symmetry on “Knot Theory and its Applications”. We look forward to receiving your contribution by 31 August, 2017. The topic of this Special Issue is focused on applications of the theory of knots to natural sciences, applications to other areas of mathematics, and relationships between science, mathematics, and the theory of knots. Knot theory is a relatively recent mathematical subject. It began in earnest at the end of the nineteenth century, with tabulations of knots made by mathematics, such as Peter Guthrie Tait, in response to the vortex theory of Lord Kelvin. Kelvin (Sir William Thompson) theorized that atoms were knotted vortices in the luminiferous aether. Kelvin began the first physical theories of knots in fluid flow, but his grand theory of atoms as vortices was eventually discarded along with the aether in the wake of special relativity. In the early twentieth century, the subject of topology and algebraic topolgy began to develop and along with it developed rigorous theory of knots, invariants of knots via fundamental group, and knot polynomials and relationships with three-dimensional manifolds. The development continued, and was influenced by the evolution of algebraic topology and differential topology. Things changed in the 1980s with the discovery of new relations with statistical mechanics, the Yang-Baxter equation, state summations, the Jones polynomial and other polynomial invariants of knots, relations of these invariants with quantum field theory in the work of Edward Witten, and many new structures related to Hopf algebras and Lie algebras. New mathematics of knots appeared in the 1990s with Vassiliev invariants and Khovanov homology. The present state-of-the-art has a deep mixture of physics and these many new invariants of knots and links. Starting in the 1980s, applications of knot theory to the structure of DNA (pioneered by Sumners, Ernst, Stasiak, Spengler, Cozzarelli) appeared, and many applications to polymers and protein folding are presently being investigated. Most recently, the work of Irvine and his collaborators produced experiments with knotted vortices in water, bringing the subject all the way back to the original ideas of Kelvin. The field of Knots and Applications is a deep and exciting one, for which we hope this Special Issue will give a glimpse into its structure and possibilities.

Prof. Louis Hirsch Kauffman
Guest Editor

Manuscript Submission Information

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. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short 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 thoroughly refereed through a single-blind 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 monthly 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 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • knots,
  • links,
  • virtual links,
  • knotoids,
  • knot polynomials,
  • quantum link invariants,
  • Yang-Baxeter equation,
  • state summations,
  • topological quantum field theory,
  • link homology,
  • polymers,
  • protein folding,
  • DNA knotting,
  • knotted vortices,
  • braided plasmas,
  • higher order linking,
  • graphs,
  • ribbon graphs.

Published Papers (5 papers)

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831 KiB  
Article
Knotoids, Braidoids and Applications
by Neslihan Gügümcü and Sofia Lambropoulou
Symmetry 2017, 9(12), 315; https://doi.org/10.3390/sym9120315 - 12 Dec 2017
Cited by 17 | Viewed by 3971
Abstract
This paper is an introduction to the theory of braidoids. Braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids. We introduce these objects and their topological equivalences, and we conclude with a potential application to [...] Read more.
This paper is an introduction to the theory of braidoids. Braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids. We introduce these objects and their topological equivalences, and we conclude with a potential application to the study of proteins. Full article
(This article belongs to the Special Issue Knot Theory and Its Applications)
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284 KiB  
Article
Open Gromov-Witten Invariants from the Augmentation Polynomial
by Matthew Mahowald
Symmetry 2017, 9(10), 232; https://doi.org/10.3390/sym9100232 - 17 Oct 2017
Viewed by 2982
Abstract
A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = O P 1 ( − 1 , − 1 ) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, [...] Read more.
A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = O P 1 ( − 1 , − 1 ) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds L K ⊂ X obtained from the conormal bundles of knots K ⊂ S 3 . This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For ( r , s ) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the ( 3 , 2 ) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants. Full article
(This article belongs to the Special Issue Knot Theory and Its Applications)
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448 KiB  
Article
Skein Invariants of Links and Their State Sum Models
by Louis H. Kauffman and Sofia Lambropoulou
Symmetry 2017, 9(10), 226; https://doi.org/10.3390/sym9100226 - 13 Oct 2017
Cited by 6 | Viewed by 3815
Abstract
We present the new skein invariants of classical links, H [ H ] , K [ K ] and D [ D ] , based on the invariants of links, H, K and D, denoting the regular isotopy version of the [...] Read more.
We present the new skein invariants of classical links, H [ H ] , K [ K ] and D [ D ] , based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants Θ d related to the Yokonuma–Hecke algebras and their 3-variable generalization Θ , which generalizes the Homflypt polynomial. H [ H ] is the regular isotopy counterpart of Θ . The invariants K [ K ] and D [ D ] are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally. Full article
(This article belongs to the Special Issue Knot Theory and Its Applications)
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713 KiB  
Article
On the Fibration Defined by the Field Lines of a Knotted Class of Electromagnetic Fields at a Particular Time
by Manuel Arrayás and José L. Trueba
Symmetry 2017, 9(10), 218; https://doi.org/10.3390/sym9100218 - 09 Oct 2017
Cited by 7 | Viewed by 3208
Abstract
A class of vacuum electromagnetic fields in which the field lines are knotted curves are reviewed. The class is obtained from two complex functions at a particular instant t = 0 so they inherit the topological properties of red the level curves of [...] Read more.
A class of vacuum electromagnetic fields in which the field lines are knotted curves are reviewed. The class is obtained from two complex functions at a particular instant t = 0 so they inherit the topological properties of red the level curves of these functions. We study the complete topological structure defined by the magnetic and electric field lines at t = 0 . This structure is not conserved in time in general, although it is possible to red find special cases in which the field lines are topologically equivalent for every value of t. Full article
(This article belongs to the Special Issue Knot Theory and Its Applications)
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3420 KiB  
Article
Planning of Knotting Based on Manipulation Skills with Consideration of Robot Mechanism/Motion and Its Realization by a Robot Hand System
by Yuji Yamakawa, Akio Namiki, Masatoshi Ishikawa and Makoto Shimojo
Symmetry 2017, 9(9), 194; https://doi.org/10.3390/sym9090194 - 15 Sep 2017
Cited by 4 | Viewed by 6607
Abstract
This paper demonstrates the relationship between the production process of a knot and manipulation skills. First, we define the description (rope intersections, grasp type and fixation positions) of a knot. Second, we clarify the characteristics of the manipulation skills from the viewpoint of [...] Read more.
This paper demonstrates the relationship between the production process of a knot and manipulation skills. First, we define the description (rope intersections, grasp type and fixation positions) of a knot. Second, we clarify the characteristics of the manipulation skills from the viewpoint of the knot description. Next, in order to obtain the production process of the knot, we propose an analysis method based on the structure of the knot and the characteristics of the manipulation skills. Using the proposed analysis method, we analyzed eight kinds of knots, formed with a single rope, two ropes or a single rope and an object. Finally, in order to validate the production process obtained by the proposed analysis method, we show experimental results of an overhand knot and a half hitch produced by using a robot hand system. Full article
(This article belongs to the Special Issue Knot Theory and Its Applications)
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