In the mathematical field of
knot theory
In 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 joined so it cannot be und ...
, 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
Surgery is a medical specialty that uses manual and instrumental techniques to diagnose or treat pathological conditions (e.g., trauma, disease, injury, malignancy), to alter bodily functions (e.g., malabsorption created by bariatric surgery s ...
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
* 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 invariantHOMFLY polynomial
In the mathematics, 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 ...
*K-theory invariants
* Atiyah–Patodi–Singer eta invariant
* Link invariantHopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between ''n''-spheres.
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Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map''
:\eta\colon S^ ...
See also
*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 descr ...
* Framed knot
*Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who intr ...
*Algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
*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 ...
*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 class ...
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