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Reshetikhin–Turaev Invariant
In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction. These invariants were discovered by Nicolai Reshetikhin and Vladimir Turaev in 1991, and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using quantum field theory. Overview To obtain an RT-invariant, one must first have a \Bbbk-linear ribbon category at hand. Each \Bbbk-linear ribbon category comes equipped with a diagrammatic calculus in which morphisms are represented by certain decorated framed tangle diagrams, where the initial and terminal objects are represented by the boundary components of the tangle. In this calculus, a (decorated framed) link diagram L, being a (decorated framed) tangle without boundary, represents an endomorphism of the monoidal identity (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Topology
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology. Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of one space within another such as knots and links in three-dimensional space. This bra–ket notation of kets and bras can be generalised, becoming maps of vector spaces associated with topological spaces that allow tensor products. Topological entanglement involving linking and braiding can be intuitively related to quantum entanglement. See also * Topological quantum field theory * Reshetikhin–Turaev invariant In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery ... References E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Invariants
In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement. List of invariants *Finite type invariant * Kontsevich invariant * Kashaev's invariant * Witten–Reshetikhin–Turaev invariant ( Chern–Simons) *Invariant differential operator *Rozansky–Witten invariant * Vassiliev knot invariant *Dehn invariant *LMO invariant *Turaev–Viro invariant *Dijkgraaf–Witten invariant *Reshetikhin–Turaev invariant *Tau-invariant *I-Invariant * Klein J-invariant *Quantum isotopy invariant * Ermakov–Lewis invariant *Hermitian invariant *Goussarov–Habiro theory of finite-type invariant *Linear quantum invariant (orthogonal function invariant) *Murakami–Ohtsuki TQFT * Generalized Casson invariant * Casson-Walker invariant *Khovanov–Rozansky invariant *HOMFLY polynomial *K-theory invariants *Atiyah–Patodi–Singer eta invariant * Link invar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Framed Link
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dehn Surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then ''filling''. Definitions * Given a 3-manifold M and a link L \subset M, the manifold M drilled along L is obtained by removing an open tubular neighborhood of L from M. If L = L_1\cup\dots\cup L_k , the drilled manifold has k torus boundary components T_1\cup\dots\cup T_k. The manifold ''M drilled along L'' is also known as the link complement, since if one removed the corresponding closed tubular neighborhood from M, one obtains a manifold diffeomorphic to M \setminus L. * Given a 3-manifold whose boundary is made of 2-tori T_1\cup\dots\cup T_k, we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components T_i of the original 3-manifold. There are many inequivalent way ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolai Reshetikhin
Nicolai Yuryevich Reshetikhin (russian: Николай Юрьевич Решетихин, born October 10, 1958 in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at Tsinghua University, China and a professor of mathematical physics at the University of Amsterdam (Korteweg-de Vries Institute for Mathematics). He is also a professor emeritus at the University of California, Berkeley. His research is in the fields of low-dimensional topology, representation theory, and quantum groups. His major contributions are in the theory of quantum integrable systems, in representation theory of quantum groups and in quantum topology. He and Vladimir Turaev constructed invariants of 3-manifolds which are expected to describe quantum Chern-Simons field theory introduced by Edward Witten. He earned his bachelor's degree and master's degree from Leningrad State University in 1982, and his Ph.D. from the Steklov Mathematical Institute in 1984. He gave a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Turaev
Vladimir Georgievich Turaev (Владимир Георгиевич Тураев, born in 1954) is a Russian mathematician, specializing in topology. Turaev received in 1979 from the Steklov Institute of Mathematics his Candidate of Sciences degree (PhD) under Oleg Viro. Turaev was a professor at the University of Strasbourg and then became a professor at Indiana University. In 2016 he was elected a Fellow of the American Mathematical Society. Turaev's research deals with low-dimensional topology, quantum topology, and knot theory and their interconnections with quantum field theory. In 1991 Reshetikhin and Turaev published a mathematical construction of new topological invariants of compact oriented 3-manifolds and framed links in these manifolds, corresponding to a mathematical implementation of ideas in quantum field theory published by Witten; the invariants are now called Witten-Reshetikhin-Turaev (or Reshetikhin-Turaev) invariants. In 1992 Turaev and Viro introduced a new fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribbon Category
In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category. Definition A monoidal category \mathcal C is, loosely speaking, a category equipped with a notion resembling the tensor product (of vector spaces, say). That is, for any two objects C_1, C_2 \in \mathcal C, there is an object C_1 \otimes C_2 \in \mathcal C. The assignment C_1, C_2 \mapsto C_1 \otimes C_2 is supposed to be functorial and needs to require a number of further properties such as a unit object 1 and an associativity isomorphism. Such a category is called braided if there are isomorphisms :c_: C_1 \otimes C_2 \stackrel \cong \rightarrow C_2 \otimes C_1. A braided monoidal category is called a ribbon category if the category is left rigid and has a family of ''twists''. The former means that for each object C there is another object (called the left dual), C^*, with maps :1 \rightarrow C \otimes C^*, C^* \otimes C \rightarrow 1 such that the compositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangle Diagram
In mathematics, a tangle is generally one of two related concepts: * In John Conway's definition, an ''n''-tangle is a proper embedding of the disjoint union of ''n'' arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2''n'' marked points on the ball's boundary. * In link theory, a tangle is an embedding of ''n'' arcs and ''m'' circles into \mathbf^2 \times ,1/math> – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance. (A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, '' Journal of Combinatorial Theory'' B 52 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to matroids.) The balance of this article discusses Conway's sense of tangles; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirby Moves
In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if ''M'' and ''N'' are 3-manifolds, resulting from Dehn surgery on framed links ''L'' and ''J'' respectively, then they are homeomorphic if and only if ''L'' and ''J'' are related by a sequence of Kirby moves. According to the Lickorish–Wallace theorem any closure (mathematics), closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere. Some ambiguity exists in the literature on the precise use of the term "Kirby moves". Different presentations of "Kirby calculus" have a different set of moves and these are sometimes called Kirby moves. Kirby's original formulation involved two kinds of move, the "blow-up" and the "handle slide"; Roger Fenn and Colin P. Rourke, Colin Rourke exhibited an equivalent construction in term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribbon Hopf Algebra
A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, cotton, and jute and of synthetic materials, such as polyester, nylon, and polypropylene. Ribbon is used for useful, ornamental, and symbolic purposes. Cultures around the world use ribbon in their hair, around the body, and as ornament on non-human animals, buildings, and packaging. Some popular fabrics used to make ribbons are satin, organza, sheer, silk, velvet, and grosgrain. Etymology The word ribbon comes from Middle English ''ribban'' or ''riban'' from Old French ''ruban'', which is probably of Germanic origin. Cloth Along with that of tapes, fringes, and other smallwares, the manufacture of cloth ribbons forms a special department of the textile industries. The essential feature of a ribbon loom is the simultaneous weaving in one loom frame of two or more we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
