In the mathematical field of
knot theory
In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
, a quantum knot invariant or quantum invariant of a
knot
A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...
or link is a linear sum of
colored Jones polynomial of
surgery presentations of the
knot complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
.
List of invariants
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Finite type invariant In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be described) to an invariant of certain singular ...
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Kontsevich invariant
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Kashaev's invariant
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Witten–Reshetikhin–Turaev invariant (
Chern–Simons)
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Invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fu ...
*Rozansky–Witten invariant
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Vassiliev knot invariant In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be described) to an invariant of certain singular ...
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Dehn invariant
*LMO invariant
*Turaev–Viro invariant
*Dijkgraaf–Witten invariant
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Reshetikhin–Turaev invariant
*Tau-invariant
*I-Invariant
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Klein J-invariant
In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is ho ...
*Quantum isotopy invariant
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Ermakov–Lewis invariant Many quantum mechanical Hamiltonians are time dependent. Methods to solve problems where there is an explicit time dependence is an open subject nowadays. It is important to look for constants of motion or invariants for problems of this kind. For ...
*Hermitian invariant
*Goussarov–Habiro theory of finite-type invariant
*Linear quantum invariant (orthogonal function invariant)
*Murakami–Ohtsuki
TQFT
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Generalized Casson invariant In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.
Kevin Walker (1992) found an extension to ...
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Casson-Walker invariant In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.
Kevin Walker (1992) found an extension to ...
*Khovanov–Rozansky invariant
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HOMFLY polynomial
In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables ''m'' and ...
*K-theory invariants
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Atiyah–Patodi–Singer eta invariant
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Link invariant
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. So ...
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Casson invariant In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.
Kevin Walker (1992) found an extension to ...
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Seiberg–Witten invariants
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Gromov–Witten invariant
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 man ...
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Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf ...
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Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map''
:\eta\colon S^3 \t ...
See also
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Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
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Framed knot
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 ...
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Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and Ja ...
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Algebraic geometry
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Seifert surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link.
Such surfaces can be used to study the properties of the associated knot or link. For exampl ...
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Geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
References
Further reading
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External links
Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev
Invariant theory
Knot theory
{{Knottheory-stub