47 relations: Alexander polynomial, Alexander's theorem, Bracket polynomial, Braid group, Bulletin of the American Mathematical Society, Chern–Simons theory, Colin Adams (mathematician), Dror Bar-Natan, Edward Witten, Euler characteristic, Finite type invariant, Fundamental representation, Gauge theory, Hernando Burgos-Soto, HOMFLY polynomial, Hyperbolic volume, Irreducible representation, Journal of Knot Theory and Its Ramifications, Khovanov homology, Knot (mathematics), Knot complement, Knot invariant, Knot polynomial, Knot theory, Kontsevich invariant, Laurent polynomial, Link (knot theory), Louis Kauffman, Maxim Kontsevich, Mikhail Khovanov, Morwen Thistlethwaite, Palindrome, Potts model, Reidemeister move, Ring (mathematics), Root of unity, Skein relation, Statistical mechanics, Tangle (mathematics), Temperley–Lieb algebra, Unknot, Unlink, Vacuum expectation value, Vaughan Jones, Volume conjecture, Wilson loop, Writhe.
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type.
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid.
In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links.
In mathematics, the braid group on strands (denoted), also known as the Artin braid group, is the group whose elements are equivalence classes of n-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 Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry.
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.
The Chern–Simons theory, named after Shiing-Shen Chern and James Harris Simons, is a 3-dimensional topological quantum field theory of Schwarz type, developed by Edward Witten.
Colin Conrad Adams (born October 13, 1956) is a mathematician primarily working in the areas of hyperbolic 3-manifolds and knot theory.
Dror Bar-Natan (דרוֹר בָר-נָתָן; born January 30, 1966) is a Professor at the University of Toronto Department of Mathematics, Canada.
Edward Witten (born August 26, 1951) is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant, 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.
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight.
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
Hernando Burgos Soto is a Canadian (Colombian born) writer and mathematician, professor of mathematics at George Brown College.
In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot.
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric.
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.
The Journal of Knot Theory and Its Ramifications was established in 1992 by Louis Kauffman and was the first journal purely devoted to knot theory.
In mathematics, Khovanov homology is an oriented link invariant that arises as the homology of a chain complex.
In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).
In mathematics, the knot complement of a tame knot K is the three-dimensional space surrounding the knot.
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.
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
In topology, knot theory is the study of mathematical knots.
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.
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb.
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.
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.
Maxim Lvovich Kontsevich (Макси́м Льво́вич Конце́вич;; born 25 August 1964) is a Russian and French mathematician.
Mikhail Khovanov (Михаил Хованов; born 1972) is a Russian-American professor of mathematics at Columbia University.
Morwen B. Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville.
A palindrome is a word, number, or other sequence of characters which reads the same backward as forward, such as madam or racecar.
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice.
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram.
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
Skein relations are a mathematical tool used to study knots.
Statistical mechanics is one of the pillars of modern physics.
In mathematics, a tangle is generally one of two related concepts.
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb.
The unknot arises in the mathematical theory of knots.
In the mathematical field of knot theory, the unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average, expected value in the vacuum.
Sir Vaughan Frederick Randal Jones (born 31 December 1952) is a New Zealand and American mathematician, known for his work on von Neumann algebras and knot polynomials.
In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements.
In gauge theory, a Wilson loop (named after Kenneth G. Wilson) is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop.
In knot theory, there are several competing notions of the quantity writhe, or Wr.