|
Quantum Invariant
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. 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 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 *K-theory invariants * Atiyah–Patodi–Singer eta invariant * Link invar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory. In condensed matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states. Overview In a topological field theory, correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 which takes one knot to the other. 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 mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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^2, and proved that \eta is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles :\eta^(x),\eta^(y) \subset S^3 is equal to 1, for any x \neq y \in S^2. It was later shown that the homotopy group \pi_3(S^2) is the infinite cyclic group generated by \eta. In 1951, Jean-Pierre Serre proved that the rational homotopy groups :\pi_i(S^n) \otimes \mathbb for an odd-dimensional sphere (n odd) are zero unless i is equal to 0 or ''n''. However, for an even-dimensional sphere (''n'' even), there is one more bit of infinite cyclic homotopy in degree 2n-1. Definition Let \phi \colon S^ \to S^n be a continuous map (assume n>1). Then we can form the cell complex : C_\phi = S^n \cup_\ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arf 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. If ''F'' is a Seifert surface of a knot, then the homology group has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot. Definition by Seifert matrix Let V = v_ be a Seifert matrix of the knot, constructed from a set of curves on a Seifert surface of genus ''g'' which represent a basis for the first homology of the surface. This means that ''V'' is a matrix with the property that is a symplectic matrix. The ''Arf invariant'' of the knot is the residue of :\sum\limits^g_ v_ v_ \pmod 2. Specifically, if \, i = 1 \ldots g, is a symplectic basis for the intersection form on the Seifert surface, then :\operatorname(K) = \sum\limi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important. Definition Consider the following: *''X'': a closed symplectic manifold of dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 theory. Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tends to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory. For detailed descriptions of Seiberg–Witten invariants see , , , , . For the relation to symplectic manifolds and Gromov–Witten invariants see . For the early history see . Spin''c''-structures The Spin''c'' group (in dimension 4) is :(U(1) \times \mathrm(4))/(\Z/2\Z). where the \Z/2\Z acts as a sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop (1995) extended the invariant to all closed oriented 3-manifolds. Definition A Casson invariant is a surjective map λ from oriented integral homology 3-spheres to Z satisfying the following properties: *λ(S3) = 0. *Let Σ be an integral homology 3-sphere. Then for any knot ''K'' and for any integer ''n'', the difference ::\lambda\left(\Sigma+\frac\cdot K\right)-\lambda\left(\Sigma+\frac\cdot K\right) :is independent of ''n''. Here \Sigma+\frac\cdot K denotes \frac Dehn surgery on Σ by ''K''. *For any boundary link ''K'' ∪ ''L'' in Σ the following expression is zero: ::\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) -\lambda\left(\Sigma+\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot 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. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a ''knot invariant'' is a rule that assigns to any knot a quantity such that if and are equivalent then ."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,Purcell, Jessica (2020). ''Hyperbolic Knot Theory'', p.7. American Mathematical Society. "A ''knot invariant'' is a function from the set of knots to some other set whose value depends only on the equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 defined using zeta function regularization. It was introduced by who used it to extend the Hirzebruch signature theorem to manifolds with boundary. The name comes from the fact that it is a generalization of the Dirichlet eta function. They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold. defined the signature defect of the boundary of a manifold as the eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at ''s''=0 or 1 of a Shimizu L-function. Definition The eta invariant of self-adjoint operator ''A'' is given by ''η''''A''(0), where ''η'' is the ana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> TQFT")
