We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in 1. In particular, following the theory of quantum invariants we work with ...‘rotational’ virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.
•Description of chord diagrams in the setting of rotational virtual knots.•Construction of Lie algebra weight systems of rotational virtual knots.•Generalization of quantum invariants of rotational virtual knots.
The computer artificial intelligence system AlphaFold has recently predicted previously unknown three‐dimensional structures of thousands of proteins. Focusing on the subset with high‐confidence ...scores, we algorithmically analyze these predictions for cases where the protein backbone exhibits rare topological complexity, that is, knotting. Amongst others, we discovered a 71‐knot, the most topologically complex knot ever found in a protein, as well several six‐crossing composite knots comprised of two methyltransferase or carbonic anhydrase domains, each containing a simple trefoil knot. These deeply embedded composite knots occur evidently by gene duplication and interconnection of knotted dimers. Finally, we report two new five‐crossing knots including the first 51‐knot. Our list of analyzed structures forms the basis for future experimental studies to confirm these novel‐knotted topologies and to explore their complex folding mechanisms.
With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this ...area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced. This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots. Various other topics are then discussed, such as Gauss diagram formulae, before the book ends with Vassiliev's original construction.
The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory
An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the ...field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra.
The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f -polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.
Knots and crossings Virtual knots and links CURVES IN THE PLANE VIRTUAL LINKS ORIENTED VIRTUAL LINK DIAGRAMS
Linking invariants CONDITIONAL STATEMENTS WRITHE AND LINKING NUMBER DIFFERENCE NUMBER CROSSING WEIGHT NUMBERS
A multiverse of knots FLAT AND FREE LINKS WELDED, SINGULAR, AND PSEUDO KNOTS NEW KNOT THEORIES
Crossing invariants CROSSING NUMBERS UNKNOTTING NUMBERS UNKNOTTING SEQUENCE NUMBERS
Constructing knots SYMMETRY TANGLES, MUTATION, AND PERIODIC LINKS PERIODIC LINKS AND SATELLITE KNOTS
Knot polynomials The bracket polynomial THE NORMALIZED KAUFFMAN BRACKET POLYNOMIAL THE STATE SUM THE IMAGE OF THE F -POLYNOMIAL
Surfaces SURFACES CONSTRUCTIONS OF VIRTUAL LINKS GENUS OF A VIRTUAL LINK
Bracket polynomial II STATES AND THE BOUNDARY PROPERTY PROPER STATES DIAGRAMS WITH ONE VIRTUAL CROSSING
The checkerboard framing CHECKERBOARD FRAMINGS CUT POINTS EXTENDING THE KAUFFMAN-MURASUGI-THISTLETHWAITE THEOREM
Modifications of the bracket polynomial THE FLAT BRACKET THE ARROW POLYNOMIAL VASSILIEV INVARIANTS
Algebraic structures Quandles TRICOLORING QUANDLES KNOT QUANDLES
Knots and quandles A LITTLE LINEAR ALGEBRA AND THE TREFOIL THE DETERMINANT OF A KNOT THE ALEXANDER POLYNOMIAL THE FUNDAMENTAL GROUP
Biquandles THE BIQUANDLE STRUCTURE THE GENERALIZED ALEXANDER POLYNOMIAL
Gauss diagrams GAUSS WORDS AND DIAGRAMS PARITY AND PARITY INVARIANTS CROSSING WEIGHT NUMBER
Applications QUANTUM COMPUTATION TEXTILES
Appendix A: Tables Appendix B: References by Chapter
Open problems and projects appear at the end of each chapter.
"This text provides an excellent entry point into virtual knot theory for undergraduates. Beginning with few prerequisites, the reader will advance to master the combinatorial and algebraic techniques that are most often employed in the literature. A student-centered book on the multiverse of knots (i.e., virtual knots, flat knots, free knots, welded knots, and pseudo knots) has long been awaited. The text aims not only to advertise recent developments in the field but to bring students to a point where they can begin thinking about interesting problems on their own. Each chapter contains not only exercises but projects, lists of open problems, and a carefully curated reading list. Students preparing to embark on an undergraduate research project in knot theory or virtual knot theory will greatly benefit from reading this well-written book!" —Micah Chrisman, Ph.D., Associate Professor, Monmouth University
"This book will be greatly helpful and perfect for undergraduate and graduate students to study knot theory and see how ideas and techniques of mathematics learned at colleges or universities are used in research. Virtual knots are a hot topic in knot theory. By comparing virtual with classical, the book enables readers to understand the essence more easily and clearly." —Seiichi Kamada, Vice-Director of Osaka City University Advanced Mathematical Institute and Professor of Mathematics, Osaka City University
"This is an excellent and well-organized introduction to classical and virtual knot theory that makes these subjects accessible to interested persons who may be unacquainted with point set topology or algebraic topology. The prerequisites for reading the book are a familiarity with basic college algebra and then later some abstract algebra and a familiarity or willingness to work with graphs (in the sense of graph theory) and pictorial diagrams (for knots and links) that are related to graphs. With this much background the book develops related topological themes such as knot polynomials, surfaces and quandles in a self-contained and clear manner. The subject of virtual knot theory is relatively new, having been introduced by Kauffman and by Goussarov, Polyak and Viro around 1996. Virtual knot theory can be learned right along with classical knot theory, as this book demonstrates, and it is a current research topic as well. So this book, elementary as it is, brings the reader right up to the frontier of present work in the theory of knots. It is exciting that knot theory, like graph theory, affords this possibility of stepping forward into the creative unknown." —Louis H. Kauffman, Professor of Mathematics, University of Illinois at Chicago
Heather A. Dye is an associate professor of mathematics at McKendree University in Lebanon, Illinois, where she teaches linear algebra, probability, graph theory, and knot theory. She has published articles on virtual knot theory in the Journal of Knot Theory and its Ramifications, Algebraic and Geometric Topology, and Topology and its Applications . She is a member of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA).
Molecular Knots Fielden, Stephen D. P.; Leigh, David A.; Woltering, Steffen L.
Angewandte Chemie (International ed.),
September 4, 2017, Letnik:
56, Številka:
37
Journal Article
Recenzirano
Odprti dostop
The first synthetic molecular trefoil knot was prepared in the late 1980s. However, it is only in the last few years that more complex small‐molecule knot topologies have been realized through ...chemical synthesis. The steric restrictions imposed on molecular strands by knotting can impart significant physical and chemical properties, including chirality, strong and selective ion binding, and catalytic activity. As the number and complexity of accessible molecular knot topologies increases, it will become increasingly useful for chemists to adopt the knot terminology employed by other disciplines. Here we give an overview of synthetic strategies towards molecular knots and outline the principles of knot, braid, and tangle theory appropriate to chemistry and molecular structure.
Thou shalt knot … This Review gives an overview of the field of closed‐loop molecular entanglements in terms of their synthesis, and outlines the principles of knot, braid, and tangle theory appropriate to chemistry and molecular structure. The steric restrictions imposed on molecular strands by knotting can impart significant physical and chemical properties, including chirality, strong and selective ion binding, and catalytic activity.
Optical knots and links have attracted great attention because of their exotic topological characteristics. Recent investigations have shown that the information encoding based on optical knots could ...possess robust features against external perturbations. However, as a superior coding scheme, it is also necessary to achieve a high capacity, which is hard to be fulfilled by existing knot-carriers owing to the limit number of associated topological invariants. Thus, how to realize the knot-based information coding with a high capacity is a key problem to be solved. Here, we create a type of nested vortex knot, and show that it can be used to fulfill the robust information coding with a high capacity assisted by a large number of intrinsic topological invariants. In experiments, we design and fabricate metasurface holograms to generate light fields sustaining different kinds of nested vortex links. Furthermore, we verify the feasibility of the high-capacity coding scheme based on those topological optical knots. Our work opens another way to realize the robust and high-capacity optical coding, which may have useful impacts on the field of information transfer and storage.
Untying knots with force Li, Pan T. X.
Nature chemical biology,
09/2021, Letnik:
17, Številka:
9
Journal Article
Recenzirano
RNA knot-like structures function as an efficient physical barrier to RNA exoribonucleases. Single-molecule mechanical manipulation is used to unfold these structures and unravel the cause of their ...unusual mechanical resistance from a different direction.
Topological descriptions of protein folding Flapan, Erica; He, Adam; Wong, Helen
Proceedings of the National Academy of Sciences - PNAS,
05/2019, Letnik:
116, Številka:
19
Journal Article
Recenzirano
Odprti dostop
How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to ...investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. Bölinger D, et al. (2010) PLoS Comput Biol 6:e1000731 for the folding of the 6₁-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor’s twisted hairpin theory of knot folding for the 3₁-knotted proteins and the 4₁-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the 4₁-knotted phytochromes and the 5₂-and 6₁-knotted proteins.
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