In the mathematical field of
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), 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 ...
, a quantum knot invariant or quantum invariant of a
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
or link is a linear sum of
colored Jones polynomial of
surgery
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...
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 In the knot theory, mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral of an oriented framed link (knot theory), link, is a Finite type invariant#The universal Vassiliev invariant, universal Vassiliev invar ...
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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 fun ...
*Rozansky–Witten invariant
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Vassiliev knot invariant
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Dehn invariant
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissection problem, dissected") into another, and whether a polyhedron or its dissections can Honeycomb (geometry), tile s ...
*LMO invariant
*Turaev–Viro invariant
*Dijkgraaf–Witten invariant
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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 ...
*Tau-invariant
*I-Invariant
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Klein J-invariant
*Quantum isotopy invariant
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Ermakov–Lewis invariant
*Hermitian invariant
*Goussarov–Habiro theory of finite-type invariant
*Linear quantum invariant (orthogonal function invariant)
*Murakami–Ohtsuki
TQFT
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathem ...
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Generalized Casson invariant
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Casson-Walker invariant
*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 In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are def ...
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Link invariant
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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
In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by , using the Seiberg–Witten theory studied by during their investigations of Seiberg–Witten gauge theo ...
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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 i ...
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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 \to S^ ...
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 Jam ...
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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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 example ...
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Geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...
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