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Bracket Polynomial
In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister moves), a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds. The bracket polynomial was discovered by Louis Kauffman in 1987. Definition The bracket polynomial of any (unoriented) link diagram L, denoted \langle L \rangle , is a polynomial in the variable A, characterized by the three rules: * \langle \bigcirc \rangle = 1 , where \bigcirc is the standard diagram of the unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the leas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Framed Link
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] |
Reidemeister Move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttingen. In 1920, he got the staatsexamen (master's degree) in mathematics, philosophy, physics, chemistry, and geology. He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. He became interested in differential geometry; he edited Wilhelm Blaschke's 2nd volume about that issue, and both made an acclaimed contribution to the Jena DMV conference in Sep 1921. In October 1922 (or 1923) he was appointed assistant professor at the University of Vienna. While there he became familiar with the work of Wilhelm Wirtinger on knot theory, and became closely connected to Hans Hahn and the Vienna Circle. Its manifesto (1929) lists one of Reidemeister's publications i ... [...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 equiva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jones Polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^ with integer coefficients. Definition by the bracket Suppose we have an oriented link L, given as a knot diagram. We will define the Jones polynomial, V(L), using Louis Kauffman's bracket polynomial, which we denote by \langle~\rangle. Here the bracket polynomial is a Laurent polynomial in the variable A with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) :X(L) = (-A^3)^\langle L \rangle, where w(L) denotes the writhe of L in its given diagram. The writhe of a diagram is the number of positive crossings (L_ in the figure below) minus the number of negative crossings (L_). The writhe is not a knot invariant. X(L) is a knot invariant since it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis Kauffman
Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, topologist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial. Biography Kauffman was valedictorian of his graduating class at Norwood Norfolk Central High School in 1962. He received his B.S. at the Massachusetts Institute of Technology in 1966 and his Ph.D. in mathematics from Princeton University in 1972 (with William Browder as thesis advisor). Kauffman has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institut des Hautes Études Scientifiques in Bures Sur Yevette, France, the Institut Henri Poincaré in Paris, France, the University of Bologna, Italy, the Federal University of Pernambuco in Recife, Brazil, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot. The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to the knot sum operation. Unknotting problem Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP. It is known that knot Floer homology and Khova ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kauffman Bracket2
Kaufmann is a surname with many variants such as Kauffmann, Kaufman, and Kauffman. In German, the name means ''merchant''. It is the cognate of the English '' Chapman'' (which had a similar meaning in the Middle Ages, though it disappeared from modern English). ''Kaufmann'' may refer to: Kaufmann * Alexander Kaufmann (1817–1893), German poet and folklorist, brother of Leopold * Aloys P. Kaufmann (1902–1984), Mayor of St. Louis, Missouri * Andrea Kaufmann (born 1969), Austrian politician * Andreas Kaufmann (born 1973), German footballer * Andy Kaufmann (born 1967), American basketball player * Arthur Kaufmann (1872–1938), Austrian attorney, philosopher and chess master * Arthur Kaufmann (artist) (1888–1971, German avant-garde painter * Bob Kauffman (1946–2015), American basketball player * Carl Kaufmann (1936–2008), West German sprint runner * Christian Kaufmann (alpine guide) (1872-1939), Swiss mountain guide active in Canada * Christian Kaufmann (canoeist) ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
