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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) by 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 si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reidemeister Move
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. and, independently, , demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves. Each move operates on a small region of the diagram and is one of three types: No other part of the diagram is involved in the picture of a move, and a planar isotopy may distort the picture. The numbering for the types of moves corresponds to how many strands are involved, e.g. a type II move operates on two strands of the diagram. One important context in which the Reidemeister moves appear is in defining knot invariants. By demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves, an invariant is defined. Many important invariants can be defined in this way, including the Jones polynomial. The type I move is the only move that affects the writh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Knot
In the mathematical field of knot theory, a chiral knot is a knot that is ''not'' equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible. There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.. Background The possible chirality of certain knots was suspected since 1847 when Johann Listing asserted that the trefoil was chiral, and this was proven by Max Dehn in 1914. P. G. Tait found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had even crossing number. Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphich ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skein (HOMFLY)
Skein may refer to: * A flock of geese or ducks in flight * A wound ball of yarn with a centre pull strand; see Hank * A metal piece fitted over the end of a wagon axle, to which the wheel is mounted * Skein (unit), a unit of length used by weavers and tailors * Skein dubh, a Scottish knife * Skein module, a mathematical concept * Skein relation, a mathematical concept often used to give a simple definition of knot polynomials * Skein (comics), a fictional supervillain in the Marvel Comics universe * Skein (hash function), a candidate hash function to the NIST hash function competition from Bruce Schneier et al. See also * '' The Tangled Skein'', a novel by Baroness Orczy * '' With a Tangled Skein'', a novel by Piers Anthony, book three of ''Incarnations of Immortality'' * Skien Skien () is a municipality in Telemark county, Norway. It is located in the traditional district of Grenland, although historically it belonged to Grenmar/Skiensfjorden, while Grenland referre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skein Relation
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. However, the converse is not true. Skein relations are often used to give a simple definition of knot polynomials. A skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region. For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively. Definition A skein relationship requires three link diagrams that are identical except at one crossing. The three diagrams must exhibit the three possibilities that could occur for the two line segments at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Trace
Markov ( Bulgarian, ), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics * Ivana Markova (1938–2024), Czechoslovak-British emeritus professor of psychology at the University of Stirling *John Markoff (sociologist) (born 1942), American professor of sociology and history at the University of Pittsburgh * Konstantin Markov (1905–1980), Soviet geomorphologist and quaternary geologist Mathematics, science, and technology * Alexander V. Markov (born 1965), Russian biologist *Andrey Markov (1856–1922), Russian mathematician * Andrey Markov Jr. (1903–1979), Russian mathematician and son of Andrey Markov *Elena Vladimirovna Markova (1923–2023), Soviet and Russian cyberneticist, Doctor of Technical Sciences, gulag convict and memoirist. *John Markoff (born 1949), American journalist of computer industry and technology *Moisey Markov (1908–1994), Russian physicist * Vladimir Andreevich Markov (1871–1897), Russia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperley–Lieb Algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrix, transfer matrices, invented by Harold Neville Vazeille Temperley, Neville Temperley and Elliott H. Lieb, Elliott Lieb. It is also related to integrable models, knot theory and the braid group, braid groups, quantum groups and subfactors of von Neumann algebras. Structure Generators and relations Let R be a commutative ring and fix \delta \in R. The Temperley–Lieb algebra TL_n(\delta) is the algebra (ring theory), R-algebra generated by the elements e_1, e_2, \ldots, e_, subject to the Jones relations: *e_i^2 = \delta e_i for all 1 \leq i \leq n-1 *e_i e_ e_i = e_i for all 1 \leq i \leq n-2 *e_i e_ e_i = e_i for all 2 \leq i \leq n-1 *e_i e_j = e_j e_i for all 1 \leq i,j \leq n-1 such that , i-j, \neq 1 Using these relations, any product of generators e_i can be brought to Jones' normal form: : E= \big(e_e_\cdots e_\big)\big(e_e_\cdots e_\big)\cdots\big(e_e_\cdots e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Emil Artin, Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander's Theorem
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander II, who published a proof in 1923. Braids were first considered as a tool of knot theory In 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 und ... by Alexander. His theorem gives a positive answer to the question ''Is it always possible to transform a given knot into a closed braid?'' A good construction example is found in Colin Adams's book. However, the correspondence between knots and braids is clearly not one-to-one: a knot may have many braid representations. For example, conjugate braids yield equivalent knots. This leads to a second fund ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potts Model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively. The model is named after Renfrey Potts, who described the model near the end of his 1951 Ph.D. thesis. The model was related to the "planar Potts" or " clock model", which was suggested to him by his advisor, Cyril Domb. The four-state Potts model is sometimes known as the Ashkin–Teller model, after Julius Ashkin and Edward Teller, who considered an equivalent model in 1943. The Potts model is related to, and generalized by, several other models, including the XY model, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
