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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 knots that vanishes on singular knots with ''m'' + 1 singularities and does not vanish on some singular knot with 'm' singularities. It is then said to be of type or order m. We give the combinatorial definition of finite type invariant due to Goussarov, and (independently) Joan Birman and Xiao-Song Lin. Let ''V'' be a knot invariant. Define ''V''1 to be defined on a knot with one transverse singularity. Consider a knot ''K'' to be a smooth embedding of a circle into \R^3. Let ''K be a smooth immersion of a circle into \mathbb R^3 with one transverse double point. Then : V^1(K') = V(K_+) - V(K_-), where K_+ is obtained from ''K'' by resolving the double point by pushing up one strand above the other, and ''K_-'' is obta ... [...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] |
Topology (journal)
''Topology'' was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of ''Topology'' appeared in 2009. Pricing dispute On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published. In 2007 the former editors of ''Topology'' announced the launch of the ''Journal of Topology'', published by Oxford University Press ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willerton's Fish
In knot theory, Willerton's fish is an unexplained relationship between the first two Vassiliev invariants of a knot. These invariants are , the quadratic coefficient of the Alexander–Conway polynomial, and , an order-three invariant derived from the Jones polynomial... See in particular Section 4.2.1, "Willerton's fish and families of knots". When the values of and , for knots of a given fixed crossing number, are used as the and coordinates of a scatter plot, the points of the plot appear to fill a fish-shaped region of the plane, with a lobed body and two sharp tail fins. The region appears to be bounded by cubic curves, suggesting that the crossing number, , and may be related to each other by not-yet-proven inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * .... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutant Knot
In the mathematical field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose ''K'' is a knot given in the form of a knot diagram. Consider a disc ''D'' in the projection plane of the diagram whose boundary circle intersects ''K'' exactly four times. We may suppose that (after planar isotopy) the disc is geometrically round and the four points of intersection on its boundary with ''K'' are equally spaced. The part of the knot inside the disc is a tangle. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a mutant of ''K''. Mutants can be difficult to distinguish as they have a number of the same invariants. They have the same hyperbolic volume (by a result of Ruberman), and have the same HOMFLY polynomials. Examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in 2012, and the Breakthrough Prize in Mathematics in 2014. Academic career and research He was born into the family of Lev Kontsevich, Soviet orientalist and author of the Kontsevich system. After ranking second in the All-Union Mathematics Olympiads, he attended Moscow State University but left without a degree in 1985 to become a researcher at the Institute for Information Transmission Problems in Moscow. While at the institute he published papers that caught the interest of the Max Planck Institute in Bonn and was invited for three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oleg Viro
Oleg Yanovich Viro (russian: Олег Янович Виро) (b. 13 May 1948, Leningrad, USSR) is a Russian mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory. Contributions Viro developed a "patchworking" technique in algebraic geometry, which allows real algebraic varieties to be constructed by a "cut and paste" method. Using this technique, Viro completed the isotopy classification of non-singular plane projective curves of degree 7. The patchworking technique was one of the fundamental ideas which motivated the development of tropical geometry. In topology, Viro is most known for his joint work with Vladimir Turaev, in which the Turaev-Viro invariants (relatives of the Reshetikhin-Turaev invariants) and related topological quantum field theory notions were introduced. Education and career Viro studied at the Leningrad State University where he received his Ph.D. degree in 1974; his advisor w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Link
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a ''trivial'' reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. For example, a co-dimension 2 link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles. The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form a Brunnian link and in fact constitute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Anatolyevich Vassiliev
Victor Anatolyevich Vassiliev or Vasilyev ( ru , Виктор Анатольевич Васильев; born April 10, 1956), is a Soviet and Russian mathematician. He is best known for his discovery of the Vassiliev invariants in knot theory (also known as finite type invariants), which subsume many previously discovered polynomial knot invariants such as the Jones polynomial. He also works on singularity theory, topology, computational complexity theory, integral geometry, symplectic geometry, partial differential equations (geometry of wavefronts), complex analysis, combinatorics, and Picard–Lefschetz theory. Biography Vassiliev studied at the Faculty of Mathematics and Mechanics at the Lomonosov University in Moscow until 1981. From 1981 to 1987 he was Senior Researcher at the Documents and Archives Research Institute, Moscow and a part-time mathematics teacher at Specialized Mathematical School No. 57, Moscow. In 1982 he defended his Kandidat nauk thesis under Vladimir A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
