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# Knot theory

In topology, knot theory is the study of mathematical knots. 

107 relations: Alexander polynomial, Alexandre-Théophile Vandermonde, Algorithm, Alternating knot, Ambient isotopy, Analysis of algorithms, Andrew Ranicki, Annals of Mathematics, Associative property, Atom, Berkeley, California, Book of Kells, Borromean rings, Braid theory, Carl Friedrich Gauss, Celtic Christianity, Celtic knot, Chinese knotting, Chiral knot, Chirality (chemistry), Circle, Commutative property, Continuous function, Crossing number (knot theory), Curve, Digon, Dimension, DNA, Dowker notation, Edward Witten, Embedding, Endless knot, Euclidean space, Finite type invariant, Floer homology, Fundamental group, Geodesic, Geometrization conjecture, Homeomorphism, Homology (mathematics), Hopf link, Horosphere, Hyperbolic geometry, Hyperbolic link, Immersion (mathematics), Injective function, Integer, James Waddell Alexander II, John Horton Conway, Jones polynomial, ... Expand index (57 more) »

## Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type.

## Alexandre-Théophile Vandermonde

Alexandre-Théophile Vandermonde (28 February 1735 – 1 January 1796) was a French mathematician, musician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Alternating knot

In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link.

## Ambient isotopy

In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold.

## Analysis of algorithms

In computer science, the analysis of algorithms is the determination of the computational complexity of algorithms, that is the amount of time, storage and/or other resources necessary to execute them.

## Andrew Ranicki

Andrew Alexander Ranicki (born Andrzej Aleksander Ranicki; 30 December 1948 – 20 February 2018) was a British mathematician who worked on algebraic topology.

## Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Atom

An atom is the smallest constituent unit of ordinary matter that has the properties of a chemical element.

## Berkeley, California

Berkeley is a city on the east shore of San Francisco Bay in northern Alameda County, California.

## Book of Kells

The Book of Kells (Codex Cenannensis; Leabhar Cheanannais; Dublin, Trinity College Library, MS A. I., sometimes known as the Book of Columba) is an illuminated manuscript Gospel book in Latin, containing the four Gospels of the New Testament together with various prefatory texts and tables.

## Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings).

## Braid theory

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Celtic Christianity

Celtic Christianity or Insular Christianity refers broadly to certain features of Christianity that were common, or held to be common, across the Celtic-speaking world during the Early Middle Ages.

## Celtic knot

Celtic knots, called Icovellavna, (snaidhm Cheilteach, cwlwm Celtaidd) are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular art.

## Chinese knotting

Chinese knotting is a decorative handicraft art that began as a form of Chinese folk art in the Tang and Song dynasty (960–1279 CE) in China.

## Chiral knot

In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image.

## Chirality (chemistry)

Chirality is a geometric property of some molecules and ions.

## Circle

A circle is a simple closed shape.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Crossing number (knot theory)

In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

## Digon

In geometry, a digon is a polygon with two sides (edges) and two vertices.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## DNA

Deoxyribonucleic acid (DNA) is a thread-like chain of nucleotides carrying the genetic instructions used in the growth, development, functioning and reproduction of all known living organisms and many viruses.

## Dowker notation

In the mathematical field of knot theory, the Dowker notation, also called the Dowker&ndash;Thistlethwaite notation or code, for a knot is a sequence of even integers.

## Edward Witten

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.

## Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

## Endless knot

The endless knot or eternal knot (śrīvatsa; Tibetan དཔལ་བེའུ། dpal be'u; Mongolian Ulzii) is a symbolic knot and one of the Eight Auspicious Symbols.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Finite type invariant

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.

## Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.

## Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

## Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

## Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component.

## Horosphere

In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space.

## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry.

## Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others.

## John Horton Conway

John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

## Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.

## Knot

A knot is a method of fastening or securing linear material such as rope by tying or interweaving.

## Knot (mathematics)

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).

## Knot complement

In mathematics, the knot complement of a tame knot K is the three-dimensional space surrounding the knot.

## Knot group

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space.

## 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.

## Knot polynomial

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.

## Knot theory

In topology, knot theory is the study of mathematical knots.

## Kurt Reidemeister

Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space.

## List of knot theory topics

Knot theory is the study of mathematical knots.

## List of prime knots

In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum.

## Luminiferous aether

In the late 19th century, luminiferous aether or ether ("luminiferous", meaning "light-bearing"), was the postulated medium for the propagation of light.

## Max Dehn

Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German-born American mathematician and student of David Hilbert.

## Maxim Kontsevich

Maxim Lvovich Kontsevich (Макси́м Льво́вич Конце́вич;; born 25 August 1964) is a Russian and French mathematician.

## Molecular knot

In chemistry, a molecular knot, or knotane, is a mechanically-interlocked molecular architecture that is analogous to a macroscopic knot.

## N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

## Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

## Perko pair

In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot.

## Peter Tait (physicist)

Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics.

## Physical knot theory

Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics (Kauffman 1991).

## Piecewise linear manifold

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Prime knot

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Proceedings of the American Mathematical Society

Proceedings of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.

## Quantum field theory

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

## Quantum group

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebras with additional structure.

## 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.

## Quantum knots

Quantum knots is a branch of quantum mechanics that connects quantum computing with Knot theory.

## Quantum topology

Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.

## Quipu

Quipu (also spelled khipu) or talking knots, were recording devices fashioned from strings historically used by a number of cultures, particularly in the region of Andean South America.

## Real algebraic geometry

In mathematics, real algebraic geometry is the study of real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings).

## Ribbon theory

Ribbon theory is a strand of mathematics within topology that has seen particular application as regards DNA.

## Skein relation

Skein relations are a mathematical tool used to study knots.

## SnapPea

SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds.

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Statistical mechanics

Statistical mechanics is one of the pillars of modern physics.

## Tait conjectures

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.

## Tangle (mathematics)

In mathematics, a tangle is generally one of two related concepts.

## The Mathematical Intelligencer

The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals.

## Tibetan Buddhism

Tibetan Buddhism is the form of Buddhist doctrine and institutions named after the lands of Tibet, but also found in the regions surrounding the Himalayas and much of Central Asia.

## Topoisomerase

Topoisomerases are enzymes that participate in the overwinding or underwinding of DNA.

## Topological quantum computer

A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions).

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Torus knot

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3.

## Trefoil knot

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot.

## Tricolorability

In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules.

## Twist knot

In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together.

## Unknot

The unknot arises in the mathematical theory of knots.

## Unknotting problem

In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram.

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.

## Vaughan Jones

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.

## William Thomson, 1st Baron Kelvin

William Thomson, 1st Baron Kelvin, (26 June 1824 – 17 December 1907) was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824.

## William Thurston

William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.

## Wolfgang Haken

Wolfgang Haken (born June 21, 1928 in Berlin, Germany) is a mathematician who specializes in topology, in particular 3-manifolds.

## 3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

## References

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