Table of Contents
128 relations: Alexander polynomial, Alexandre-Théophile Vandermonde, Algorithm, Alternating knot, Ambient isotopy, American Mathematical Society, Analysis of algorithms, Andrew Ranicki, Annals of Mathematics, Associative property, Berkeley, California, Book of Kells, Borromean rings, Braid group, Cambridge University Press, Carl Friedrich Gauss, Celtic Christianity, Celtic knot, Charles Newton Little, Chinese knotting, Chiral knot, Chirality (chemistry), Circle, Circuit topology, Commutative property, Continuous function, Conway notation (knot theory), Crossing number (knot theory), Curve, Digon, Dimension, DNA, Dowker–Thistlethwaite notation, Edward Witten, Elsevier, Embedding, Endless knot, Euclidean space, Finite type invariant, Floer homology, Fundamental group, Gauss notation, Geodesic, Geometrization conjecture, Haken manifold, Homeomorphism, Homology (mathematics), Hopf link, Horosphere, Hyperbolic geometry, ... Expand index (78 more) »
- Low-dimensional topology
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type.
See Knot theory and Alexander polynomial
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.
See Knot theory and Alexandre-Théophile Vandermonde
Algorithm
In mathematics and computer science, an algorithm is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation.
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.
See Knot theory and Alternating knot
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.
See Knot theory and Ambient isotopy
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.
See Knot theory and American Mathematical Society
Analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them.
See Knot theory and Analysis of algorithms
Andrew Ranicki
Andrew Alexander Ranicki (born Andrzej Aleksander Ranicki; 30 December 1948 – 21 February 2018) was a British mathematician who worked on algebraic topology.
See Knot theory and Andrew Ranicki
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Knot theory and Annals of Mathematics
Associative property
In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result.
See Knot theory and Associative property
Berkeley, California
Berkeley is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States.
See Knot theory and Berkeley, 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 and Celtic Gospel book in Latin, containing the four Gospels of the New Testament together with various prefatory texts and tables.
See Knot theory and Book of Kells
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed.
See Knot theory and Borromean rings
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 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. Knot theory and braid group are low-dimensional topology.
See Knot theory and Braid group
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge.
See Knot theory and Cambridge University Press
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist who contributed to many fields in mathematics and science.
See Knot theory and Carl Friedrich Gauss
Celtic Christianity
Celtic Christianity is a form of Christianity that was common, or held to be common, across the Celtic-speaking world during the Early Middle Ages.
See Knot theory and Celtic Christianity
Celtic knot
Celtic knots (snaidhm Cheilteach, cwlwm Celtaidd, kolm Keltek, snaidhm Ceilteach) are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular art.
See Knot theory and Celtic knot
Charles Newton Little
Charles Newton Little (1858–1923) was an American mathematician and civil engineer.
See Knot theory and Charles Newton Little
Chinese knotting
Chinese knotting, also known as, is a Chinese folk art with ties to Buddhism and Taoism.
See Knot theory and Chinese knotting
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).
See Knot theory and Chiral knot
Chirality (chemistry)
In chemistry, a molecule or ion is called chiral if it cannot be superposed on its mirror image by any combination of rotations, translations, and some conformational changes.
See Knot theory and Chirality (chemistry)
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.
Circuit topology
The circuit topology of a folded linear polymer refers to the arrangement of its intra-molecular contacts.
See Knot theory and Circuit topology
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
See Knot theory and Commutative property
Continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.
See Knot theory and Continuous function
Conway notation (knot theory)
In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear.
See Knot theory and Conway notation (knot theory)
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.
See Knot theory and Crossing number (knot theory)
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Digon
In geometry, a bigon, digon, or a 2-gon, 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 polymer composed of two polynucleotide chains that coil around each other to form a double helix.
Dowker–Thistlethwaite notation
In the mathematical field of knot theory, the Dowker–Thistlethwaite (DT) notation or code, for a knot is a sequence of even integers.
See Knot theory and Dowker–Thistlethwaite notation
Edward Witten
Edward Witten (born August 26, 1951) is an American theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics.
See Knot theory and Edward Witten
Elsevier
Elsevier is a Dutch academic publishing company specializing in scientific, technical, and medical content.
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
Endless knot in a Burmese Pali manuscript The endless knot or eternal knot is a symbolic knot and one of the Eight Auspicious Symbols.
See Knot theory and Endless knot
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space.
See Knot theory and Euclidean space
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.
See Knot theory and Finite type invariant
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.
See Knot theory and Floer homology
Fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space.
See Knot theory and Fundamental group
Gauss notation
Gauss notation (also known as a Gauss code or Gauss words) is a notation for mathematical knots.
See Knot theory and Gauss notation
Geodesic
In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold.
Geometrization conjecture
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it.
See Knot theory and Geometrization conjecture
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface.
See Knot theory and Haken manifold
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
See Knot theory and Homeomorphism
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces.
See Knot theory and Homology (mathematics)
Hopf link
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.
See Knot theory and Horosphere
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry.
See Knot theory and Hyperbolic geometry
Hyperbolic link
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.
See Knot theory and Hyperbolic link
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective.
See Knot theory and Immersion (mathematics)
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.
See Knot theory and Injective function
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.
James Waddell Alexander II
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.
See Knot theory and James Waddell Alexander II
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
See Knot theory and John Horton Conway
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.
See Knot theory and Jones polynomial
Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman.
See Knot theory and Kauffman polynomial
Knot
A knot is an intentional complication in cordage which may be practical or decorative, or both.
Knot (mathematics)
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.
See Knot theory and Knot (mathematics)
Knot complement
In mathematics, the knot complement of a tame knot K is the space where the knot is not.
See Knot theory and Knot complement
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space.
See Knot theory and Knot group
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.
See Knot theory and Knot invariant
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.
See Knot theory and Knot polynomial
Knot tabulation
Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots.
See Knot theory and Knot tabulation
Knot theory
In topology, knot theory is the study of mathematical knots. Knot theory and knot theory are low-dimensional topology.
See Knot theory and Knot theory
Kurt Reidemeister
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
See Knot theory and Kurt Reidemeister
Lamp cord trick
In topology, a branch of mathematics, and specifically knot theory, the lamp cord trick is an observation that two certain spaces are homeomorphic, even if one of the components is knotted.
See Knot theory and Lamp cord trick
Link (knot theory)
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.
See Knot theory and Link (knot theory)
Linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space.
See Knot theory and Linking number
List of knot theory topics
Knot theory is the study of mathematical knots.
See Knot theory and List of knot theory topics
List of prime knots
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum.
See Knot theory and List of prime knots
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast.
See Knot theory and Lord Kelvin
Marc Lackenby
Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory.
See Knot theory and Marc Lackenby
Mathematische Zeitschrift
Mathematische Zeitschrift (German for Mathematical Journal) is a mathematical journal for pure and applied mathematics published by Springer Verlag.
See Knot theory and Mathematische Zeitschrift
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory.
Maxim Kontsevich
Maxim Lvovich Kontsevich (Макси́м Льво́вич Конце́вич,; born 25 August 1964) is a Russian and French mathematician and mathematical physicist.
See Knot theory and Maxim Kontsevich
Molecular knot
In chemistry, a molecular knot is a mechanically interlocked molecular architecture that is analogous to a macroscopic knot.
See Knot theory and Molecular knot
N-sphere
In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer.
Natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.
See Knot theory and Natural number
Orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise".
See Knot theory and Orientability
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.
See Knot theory and Perko pair
Petal projection
In knot theory, a petal projection of a knot is a knot diagram with a single crossing, at which an odd number of non-nested arcs ("petals") all meet.
See Knot theory and Petal projection
Peter Guthrie Tait
Peter Guthrie Tait (28 April 18314 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics.
See Knot theory and Peter Guthrie Tait
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).
See Knot theory and Physical knot theory
Piecewise linear manifold
In mathematics, a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it.
See Knot theory and Piecewise linear manifold
Polynomial
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.
See Knot theory and Polynomial
Prime knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.
See Knot theory and Prime knot
Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.
See Knot theory and Prime number
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University.
See Knot theory and Princeton University Press
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.
See Knot theory and Proceedings of the American Mathematical Society
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics.
See Knot theory and Quantum field theory
Quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure.
See Knot theory and Quantum group
Quantum topology
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.
See Knot theory and Quantum topology
Quipu
Quipu (also spelled khipu) are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America.
Real algebraic geometry
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings).
See Knot theory and Real algebraic geometry
Ribbon (mathematics)
In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector.
See Knot theory and Ribbon (mathematics)
Ribbon knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ribbon singularities.
See Knot theory and Ribbon knot
Scientific American
Scientific American, informally abbreviated SciAm or sometimes SA, is an American popular science magazine.
See Knot theory and Scientific American
Skein relation
Skein relations are a mathematical tool used to study knots.
See Knot theory and Skein relation
Slice knot
A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space.
See Knot theory and Slice knot
SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds.
Sphere
A sphere (from Greek) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities.
See Knot theory and Statistical mechanics
Tait conjectures
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.
See Knot theory and Tait conjectures
Tangle (mathematics)
In mathematics, a tangle is generally one of two related concepts.
See Knot theory and Tangle (mathematics)
The Mathematical Intelligencer
The Mathematical Intelligencer is a mathematical journal published by Springer Science+Business Media that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals.
See Knot theory and The Mathematical Intelligencer
Tibetan Buddhism
Tibetan Buddhism is a form of Buddhism practiced in Tibet, Bhutan and Mongolia.
See Knot theory and Tibetan Buddhism
Time complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm.
See Knot theory and Time complexity
Topoisomerase
DNA topoisomerases (or topoisomerases) are enzymes that catalyze changes in the topological state of DNA, interconverting relaxed and supercoiled forms, linked (catenated) and unlinked species, and knotted and unknotted DNA.
See Knot theory and Topoisomerase
Topological quantum computer
A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997.
See Knot theory and Topological quantum computer
Topology
Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Topology (journal)
Topology was a peer-reviewed mathematical journal covering topology and geometry.
See Knot theory and Topology (journal)
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.
See Knot theory and Torus knot
Trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot.
See Knot theory and Trefoil 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.
See Knot theory and Tricolorability
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.
See Knot theory and Twist knot
University of Oxford
The University of Oxford is a collegiate research university in Oxford, England.
See Knot theory and University of Oxford
Unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all 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.
See Knot theory and Unknotting problem
Unlink
In the mathematical field of knot theory, an 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 (31 December 19526 September 2020) was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials.
See Knot theory and Vaughan Jones
Vortex theory of the atom
The vortex theory of the atom was a 19th-century attempt by William Thomson (later Lord Kelvin) to explain why the atoms recently discovered by chemists came in only relatively few varieties but in very great numbers of each kind.
See Knot theory and Vortex theory of the atom
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.
See Knot theory and William Thurston
Wolfgang Haken
Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds.
See Knot theory and Wolfgang Haken
Wolfram Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages.
See Knot theory and Wolfram Mathematica
Wolfram Research
Wolfram Research, Inc. is an American multinational company that creates computational technology.
See Knot theory and Wolfram Research
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. Knot theory and 3-manifold are low-dimensional topology.
See Knot theory and 3-manifold
See also
Low-dimensional topology
- 3-manifold
- 4-manifold
- A Guide to the Classification Theorem for Compact Surfaces
- A Topological Picturebook
- Braid group
- Braids, Links, and Mapping Class Groups
- Clasper (mathematics)
- House with two rooms
- Knot theory
- Low-dimensional topology
- Smale conjecture
- Subgroup distortion
References
Also known as Alexander-Briggs notation, Algebraic topology based on knots, Crossing (knot theory), Hoste-Thistlethwaite knot table, Hyperbolic invariant, Knot (topology), Knot crossing, Knot diagram, Knot equivalence, Knot equivalence problem, Knot table, Link diagram, Nugatory crossing, Reducible crossing, Removable crossing, Rolfsen knot table, Rolfsen notation, Theory of knots, Thistlethwaite link table.