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.^{ [1] }^{ [2] }^{ [3] }

- Finite type invariant
- Kontsevich invariant
- Kashaev's invariant
- Witten–Reshetikhin–Turaev invariant (Chern–Simons)
- Invariant differential operator
^{ [4] } - Rozansky–Witten invariant
- Vassiliev knot invariant
- Dehn invariant
- LMO invariant
^{ [5] } - Turaev–Viro invariant
- Dijkgraaf–Witten invariant
^{ [6] } - Reshetikhin–Turaev invariant
- Tau-invariant
- I-Invariant
- Klein J-invariant
- Quantum isotopy invariant
^{ [7] } - 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 invariant
^{ [8] } - Casson invariant
- Seiberg–Witten invariant
- Gromov–Witten invariant
- Arf invariant
- Hopf invariant

**Edward Witten** is an American mathematical and theoretical physicist. He is currently the Charles Simonyi Professor in the School of Natural Sciences at the Institute for Advanced Study. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity. He is considered to be the practical founder of M-theory.

The **Chern–Simons theory** is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered firstly by a mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.

In mathematics, the **Chern theorem** states that the Euler-Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.

In gauge theory and mathematical physics, a **topological quantum field theory** is a quantum field theory which computes topological invariants.

In the mathematical field of knot theory, the **Jones polynomial** is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients.

In mathematics, specifically in symplectic topology and algebraic geometry, **Gromov–Witten** (**GW**) **invariants** are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.

In mathematics, **Khovanov homology** is an oriented link invariant that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial.

In mathematics, **Floer homology** is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

In the mathematical theory of knots, a **finite type invariant**, or **Vassiliev invariant**, is a knot invariant that can be extended to an invariant of certain singular knots that vanishes on singular knots with *m* + 1 singularities and does not vanish on some singular knot with 'm' singularities. It is then said to be of **type** or **order m**.

The **Oswald Veblen Prize in Geometry** is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen Prize is now worth US$5000, and is awarded every three years.

**Nicolai Yuryevich Reshetikhin** is a mathematical physicist, currently a professor of mathematics at the University of California, Berkeley and a professor of mathematical physics at the University of Amsterdam. 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.

In knot theory, the **Kauffman polynomial** is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as

The **Geometry Festival** is an annual mathematics conference held in the United States.

Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early, significant stimulus in knot theory would arrive later with Sir William Thomson and his vortex theory of the atom.

In the mathematical theory of knots, the **Kontsevich invariant**, also known as the **Kontsevich integral** of an oriented framed link, is a universal Vassiliev invariant in the sense that any coefficient of the Kontsevich invariant is of a finite type, and conversely any finite type invariant can be presented as a linear combination of such coefficients. It was defined by Maxim Kontsevich.

**Victor Anatolyevich Vassiliev** or **Vasilyev**, is a Soviet and Russian mathematician. He is best known for his discovery of the Vassiliev invariants in knot theory, which subsume many previously discovered polynomial knot invariants such as the Jones polynomial. He also works on singularity theory, topology, computational complexity theory, integral geometry, symplectic geometry, partial differential equations, complex analysis, combinatorics, and Picard–Lefschetz theory.

In the branch of mathematics called knot theory, the **volume conjecture** is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements.

**Vladimir Georgievich Turaev** is a Russian mathematician, specializing in topology.

**Serguei Barannikov** is a mathematician, known for his works in algebraic topology, algebraic geometry and mathematical physics.

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.

- ↑ Reshetikhin, N. & Turaev, V. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups".
*Invent. Math.***103**(1): 547. Bibcode:1991InMat.103..547R. doi:10.1007/BF01239527. S2CID 123376541. - ↑ Kontsevich, Maxim (1993). "Vassiliev's knot invariants".
*Adv. Soviet Math*.**16**: 137. - ↑ Watanabe, Tadayuki (2007). "Knotted trivalent graphs and construction of the LMO invariant from triangulations".
*Osaka J. Math*.**44**(2): 351. Retrieved 4 December 2012. - ↑ Letzter, Gail (2004). "Invariant differential operators for quantum symmetric spaces, II". arXiv: math/0406194 .
- ↑ Sawon, Justin (2000). "Topological quantum field theory and hyperkähler geometry". arXiv: math/0009222 .
- ↑ "Data" (PDF). hal.archives-ouvertes.fr. 1999. Retrieved 2019-11-04.
- ↑
- ↑ "Invariants of 3-manifolds via link polynomials and quantum groups - Springer". doi:10.1007/BF01239527. S2CID 123376541.Cite journal requires
`|journal=`

(help)

- Freedman, Michael H. (1990).
*Topology of 4-manifolds*. Princeton, N.J: Princeton University Press. ISBN 978-0691085777. OL 2220094M. - Ohtsuki, Tomotada (December 2001).
*Quantum Invariants*. World Scientific Publishing Company. ISBN 9789810246754. OL 9195378M.

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