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# Trefoil knot

In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. [1]

1. 59 relations: Alexander polynomial, Algebraic geometry, Alternating knot, Ambient isotopy, ATV Home, Braid group, Carolingian cross, Celtic cross, Chirality (mathematics), Cinquefoil knot, Circle, Clockwise, Clover, Complex number, Conway notation (knot theory), Crossing number (knot theory), Curve, Dowker–Thistlethwaite notation, Fiber bundle, Fibered knot, Figure-eight knot (mathematics), Gordian Knot, HOMFLY polynomial, Iconography, Jones polynomial, Kauffman polynomial, Knot (mathematics), Knot complement, Knot group, Knot invariant, Knot polynomial, Knot theory, Loop (topology), M. C. Escher, Mathematics, Milnor map, Mirror image, Mjölnir, Motif (visual arts), Overhand knot, Parametric equation, Plane curve, Polynomial, Pretzel link, Prime knot, Puncture (topology), Reidemeister move, Seifert fiber space, Semicubical parabola, Signature of a knot, ... Expand index (9 more) »

4. Pretzel knots and links (mathematics)
10. Twist knots

## Alexander polynomial

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

## Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical 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. Trefoil knot and alternating knot are alternating knots and links.

## 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.

## ATV Home

ATV Home was a free-to-air Cantonese television channel in Hong Kong, owned and operated by Asia Television.

## 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. Trefoil knot and braid group are knot theory.

## Carolingian cross

The Carolingian Cross is but one variation in the vast historical imagery of Christian symbolic representations of the Crucifixion of Jesus, going back to at least the ninth century.

## Celtic cross

A Celtic cross symbol The Celtic cross is a form of Christian cross featuring a nimbus or ring that emerged in Ireland, France and Great Britain in the Early Middle Ages.

## Chirality (mathematics)

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. Trefoil knot and chirality (mathematics) are knot theory.

## Cinquefoil knot

In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. Trefoil knot and cinquefoil knot are alternating knots and links, fibered knots and links, knot theory, prime knots and links, Reversible knots and links and torus knots and links.

## 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.

## Clockwise

Two-dimensional rotation can occur in two possible directions or senses of rotation.

## Clover

Clover, also called trefoil, are plants of the genus Trifolium (from Latin tres 'three' + folium 'leaf'), consisting of about 300 species of flowering plants in the legume family Fabaceae originating in Europe.

## Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

## 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. Trefoil knot and Conway notation (knot theory) are 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.

## 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.

## Dowker–Thistlethwaite notation

In the mathematical field of knot theory, the Dowker&ndash;Thistlethwaite (DT) notation or code, for a knot is a sequence of even integers. Trefoil knot and Dowker–Thistlethwaite notation are knot theory.

## Fiber bundle

In mathematics, and particularly topology, a fiber bundle (''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure.

## Fibered knot

In knot theory, a branch of mathematics, a knot or link K in the 3-dimensional sphere S^3 is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t then the intersection of F_s and F_t is exactly K. Trefoil knot and fibered knot are fibered knots and links.

## Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. Trefoil knot and figure-eight knot (mathematics) are alternating knots and links, fibered knots and links, knot theory, prime knots and links and Twist knots.

## Gordian Knot

The cutting of the Gordian Knot is an Ancient Greek legend associated with Alexander the Great in Gordium in Phrygia, regarding a complex knot that tied an oxcart.

## HOMFLY polynomial

In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called 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. Trefoil knot and HOMFLY polynomial are knot theory.

## Iconography

Iconography, as a branch of art history, studies the identification, description and interpretation of the content of images: the subjects depicted, the particular compositions and details used to do so, and other elements that are distinct from artistic style.

## Jones polynomial

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

## Kauffman polynomial

In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. Trefoil knot and Kauffman polynomial are knot theory.

## 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.

## Knot complement

In mathematics, the knot complement of a tame knot K is the space where the knot is not. Trefoil knot and knot complement are knot theory.

## 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.

## Loop (topology)

In mathematics, a loop in a topological space is a continuous function from the unit interval to such that In other words, it is a path whose initial point is equal to its terminal point.

## M. C. Escher

Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were inspired by mathematics.

## Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

## Milnor map

In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. Trefoil knot and Milnor map are knot theory.

## Mirror image

A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface.

## Mjölnir

Mjölnir (from Old Norse Mjǫllnir) is the hammer of the thunder god Thor in Norse mythology, used both as a devastating weapon and as a divine instrument to provide blessings.

## Motif (visual arts)

In art and iconography, a motif is an element of an image.

## Overhand knot

The overhand knot is one of the most fundamental knots, and it forms the basis of many others, including the simple noose, overhand loop, angler's loop, reef knot, fisherman's knot, half hitch, and water knot.

## Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

## Plane curve

In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane.

## 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.

In the mathematical theory of knots, a pretzel link is a special kind of link. Trefoil knot and pretzel link are pretzel knots and links (mathematics).

## Prime knot

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Trefoil knot and prime knot are prime knots and links.

## Puncture (topology)

In topology, puncturing a manifold is removing a finite set of points from that manifold.

## Reidemeister move

In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram.

## Seifert fiber space

A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles.

## Semicubical parabola

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form (with) in some Cartesian coordinate system.

## Signature of a knot

The signature of a knot is a topological invariant in knot theory.

## Slice knot

A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Trefoil knot and slice knot are slice knots and links.

## Torus

In geometry, a torus (tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle.

## 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 and torus knot are fibered knots and links, knot theory and torus knots and links.

## 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. Trefoil knot and tricolorability are Tricolorable knots and links.

## Triquetra

The triquetra (from the Latin adjective triquetrus "three-cornered") is a triangular figure composed of three interlaced arcs, or (equivalently) three overlapping vesicae piscis lens shapes.

## Unknot

In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Trefoil knot and unknot are fibered knots and links, knot theory, prime knots and links, slice knots and links and torus knots and links.

## Valknut

The valknut is a symbol consisting of three interlocked triangles.

## Visual arts

The visual arts are art forms such as painting, drawing, printmaking, sculpture, ceramics, photography, video, filmmaking, comics, design, crafts, and architecture.

## 3-sphere

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere.