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We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda’s 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

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).

Effects connected with the mathematical theory of knots 1 emerge in many areas of science, from physics 2 , 3 to biology 4 . Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems 5 , where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹 2 . We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension 6 , 7 . Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems. Experiments using two coupled optical ring resonators and based on the concept of synthetic dimension reveal non-Hermitian energy band structures exhibiting topologically non-trivial knots and links.

In this paper, we present two different ways for computing the Kauffman bracket skein module of S[sup.1] ×S[sup.2] , KBSM (S[sup.1] ×S[sup.2] ), via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in S[sup.1] ×S[sup.2] . We do that by imposing on V relations coming from the braid band moves . These moves reflect isotopy in S[sup.1] ×S[sup.2] and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSM (S[sup.1] ×S[sup.2] ). We show that KBSM (S[sup.1] ×S[sup.2] ) is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing KBSM (S[sup.1] ×S[sup.2] ) via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of KBSM (S[sup.1] ×S[sup.2] ).

This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24-25, 2015, at California State University, Fullerton, CA.Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves.The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in $S^3$ and other 3-manifolds.

A bstract We revisit Dimofte-Gaiotto-Gukov’s construction of 3d gauge theories associated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d N = 2 0 theory of A 1 -type on a 3-manifold, we propose a topological criterion for SU(2) / SO(3) flavor symmetry enhancement for the u (1) symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the construction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising SU(3) symmetry enhancement which is unexpected from the 6d viewpoint.

A bstract We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n -fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n -component link inside the three-sphere S 3 . The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k ( G = U(1) k ) we give a general formula for the entanglement entropy associated to an arbitrary ( m | n − m ) partition of a generic n -component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k ) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k ). For G = SU(2) k , we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).

In this paper we introduce a method that offers a detailed overview of the entanglement of an open protein chain. Further, we present a purely topological model for classifying open protein chains by also taking into account any bridge involving the backbone. To this end, we implemented the concepts of planar knotoids and bonded knotoids. We show that the planar knotoids technique provides more refined information regarding the knottedness of a protein when compared to established methods in the literature. Moreover, we demonstrate that our topological model for bonded proteins is robust enough to distinguish all types of lassos in proteins.