Quantum invariant
Encyclopedia
In the mathematical field of knot theory
, a quantum invariant of a knot
or link is a linear sum of colored Jones polynomial of surgery
presentations of the knot complement
.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a quantum invariant of a knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . 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...
or link is a linear sum of colored Jones polynomial of surgery
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
presentations of the knot complement
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
.
List of invariants
- Finite type invariantFinite type invariantIn the mathematical theory of knots, a finite type 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...
- Kontsevich invariantKontsevich invariantIn the mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral, of an oriented framed link is the universal finite type invariant in the sense that any coefficient of the Kontsevich invariant is a finite type invariant, and any finite type invariant can be...
- Kashaev's invariant
- Witten–Reshetikhin–Turaev invariant (Chern–Simons)
- Invariant differential operator
- Rozansky–Witten invariant
- Vassiliev knot invariant
- Dehn invariantHilbert's third problemThe third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
- 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 quantum invariant
- K-theory invariants
- Atiyah–Patodi–Singer eta invariant
- Link invariantKnot invariantIn the mathematical field of knot theory, a knot invariant is a quantity 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. Some invariants are indeed numbers, but invariants can range from the...
- Casson invariantCasson invariantIn 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....
- Seiberg–Witten invariantSeiberg–Witten invariantIn mathematics, Seiberg–Witten invariants are invariants of compact smooth 4-manifolds introduced by , using the Seiberg–Witten theory studied by during their investigations of Seiberg–Witten gauge theory....
- Gromov–Witten invariant
- Arf invariantArf invariant (knot)In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface...
- Hopf invariantHopf invariantIn mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.- Motivation :In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map\eta\colon S^3 \to S^2,...
See also
- Invariant theoryInvariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
- Framed knotFramed knotIn the mathematical theory of knots, a framed knot is the extension of a tame knot to an embedding of the solid torus D2 × S1 in S3....
- Chern–Simons theory
- Algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
- Seifert surfaceSeifert surfaceIn mathematics, a Seifert surface is a 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, many knot invariants are most easily calculated using a Seifert surface...
- Geometric invariant theoryGeometric invariant theoryIn mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
Further reading
- Reference to Frank Quinn: