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Kontsevich Integral
In the mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral of an oriented framed link, is a universal Vassiliev invariant in the sense that any coefficient of the Kontsevich invariant is of a finite type, and conversely any finite type invariant can be presented as a linear combination of such coefficients. It was defined by Maxim Kontsevich. The Kontsevich invariant is a universal quantum invariant in the sense that any quantum invariant may be recovered by substituting the appropriate weight system into any Jacobi diagram. Definition The Kontsevich invariant is defined by monodromy along solutions of the Knizhnik–Zamolodchikov equations. Jacobi diagram and Chord diagram Definition Let be a circle (which is a 1-dimensional manifold). As is shown in the figure on the right, a Jacobi diagram with order is the graph with vertices, with the external circle depicted as solid line circle and with dashed lines called inner grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 joined so it cannot be undone, Unknot, 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 descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Diagram STUS
Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix * Jacobi elliptic functions, a set of doubly-periodic functions * Jacobi polynomials, a class of orthogonal polynomials * Jacobi symbol, a generalization of the Legendre symbol * Jacobi coordinates, a simplification of coordinates for an n-body system * Jacobi identity for non-associative binary operations * Jacobi's formula for the derivative of the determinant of a matrix * Jacobi triple product an identity in the theory of theta functions * Jacobi's theorem (other) (various) Other: * Jacobi Medical Center, New York * Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce * Jacobi (crater), a lunar impact crater in the so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vassiliev Invariant
Vasilyev, Vasiliev or Vassiliev or Vassiljev (russian: Васильев), or Vasilyeva or Vasilieva (feminine; russian: link=no, Васильева), is a common Russian surname that is derived from the Russian given name ''Vasiliy'' (equivalent of ''Basil'') and literally means "Vasiliy's". It may refer to: *Alexander Vasilyev (musician) (born 1969), lead singer and guitar player for the Russian rock band Splean *Alexander Vasilyev (other), multiple people *Alexander Vassiliev, Russian journalist, writer and espionage historian * Boris Vasilyev (other), multiple people * Denys Vasilyev (born 1987), Ukrainian footballer *Dimitry Vassiliev (born 1979), Russian ski jumper *Dmitry Vasilyev (biathlete) (born 1962), Soviet biathlete and Olympic champion * Dmitri Vasilyev (runner), Russian runner who participated in the 2000 Summer Olympics *Dmitri Vasilyev (director) (1900–1984), Soviet film director *Dmitri Vladimirovich Vasilyev (footballer) (born 1977), Russian in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. Connected sum at a point A connected sum of two ''m''-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangle (mathematics)
In mathematics, a tangle is generally one of two related concepts: * In John Conway's definition, an ''n''-tangle is a proper embedding of the disjoint union of ''n'' arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2''n'' marked points on the ball's boundary. * In link theory, a tangle is an embedding of ''n'' arcs and ''m'' circles into \mathbf^2 \times ,1/math> – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance. (A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, '' Journal of Combinatorial Theory'' B 52 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to matroids.) The balance of this article discusses Conway's sense of tangles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Diagram 4T4
Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix * Jacobi elliptic functions, a set of doubly-periodic functions * Jacobi polynomials, a class of orthogonal polynomials * Jacobi symbol, a generalization of the Legendre symbol * Jacobi coordinates, a simplification of coordinates for an n-body system * Jacobi identity for non-associative binary operations * Jacobi's formula for the derivative of the determinant of a matrix * Jacobi triple product an identity in the theory of theta functions * Jacobi's theorem (other) (various) Other: * Jacobi Medical Center, New York * Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce * Jacobi (crater), a lunar impact crater in the so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Diagram 4T3
Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix * Jacobi elliptic functions, a set of doubly-periodic functions * Jacobi polynomials, a class of orthogonal polynomials * Jacobi symbol, a generalization of the Legendre symbol * Jacobi coordinates, a simplification of coordinates for an n-body system * Jacobi identity for non-associative binary operations * Jacobi's formula for the derivative of the determinant of a matrix * Jacobi triple product an identity in the theory of theta functions * Jacobi's theorem (other) (various) Other: * Jacobi Medical Center, New York * Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce * Jacobi (crater), a lunar impact crater in the so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Diagram 4T2
Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix * Jacobi elliptic functions, a set of doubly-periodic functions * Jacobi polynomials, a class of orthogonal polynomials * Jacobi symbol, a generalization of the Legendre symbol * Jacobi coordinates, a simplification of coordinates for an n-body system * Jacobi identity for non-associative binary operations * Jacobi's formula for the derivative of the determinant of a matrix * Jacobi triple product an identity in the theory of theta functions * Jacobi's theorem (other) (various) Other: * Jacobi Medical Center, New York * Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce * Jacobi (crater), a lunar impact crater in the so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Diagram 4T1
Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix * Jacobi elliptic functions, a set of doubly-periodic functions * Jacobi polynomials, a class of orthogonal polynomials * Jacobi symbol, a generalization of the Legendre symbol * Jacobi coordinates, a simplification of coordinates for an n-body system * Jacobi identity for non-associative binary operations * Jacobi's formula for the derivative of the determinant of a matrix * Jacobi triple product an identity in the theory of theta functions * Jacobi's theorem (other) (various) Other: * Jacobi Medical Center, New York * Jacobi (grape), another name for the French/German wine grape Pinot Noir Précoce * Jacobi (crater), a lunar impact crater in the so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chord Diagram (mathematics)
In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, and drawing the pairs of the matching as chords of the circle. The number of different chord diagrams that may be given for a set of 2n cyclically ordered objects is the double factorial (2n-1)!!. There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other. The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross. In knot theory, a chord diagram can be used to described the sequence of crossings along the planar projection of a knot, with each point at which a crossing occurs paired with the point that crosses it. To fully describe the knot, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
