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# Trefoil knot

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

59 relations: Alexander polynomial, Algebraic geometry, Alternating knot, 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 notation, Fiber bundle, Fibered knot, Fibration, 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, Reidemeister move, Seifert fiber space, Semicubical parabola, Signature of a knot, Slice knot, ... Expand index (9 more) »

## Alexander polynomial

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

## Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

## 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.

## 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), 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.

## Carolingian cross

The Carolingian cross, or Cross of triquetras, is a Christian cross symbol formed by triquetras, associated with Emperor Charlemagne of the Holy Roman Empire.

## Celtic cross

The Celtic cross is a form of Christian cross featuring a nimbus or ring that emerged in Ireland and 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.

## 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.

## Circle

A circle is a simple closed shape.

## Clockwise

Two-dimensional rotation can occur in two possible directions.

## Clover

Clover or trefoil are common names for plants of the genus Trifolium (Latin, tres "three" + folium "leaf"), consisting of about 300 species of plants in the leguminous pea family Fabaceae.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## 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.

## 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.

## 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.

## Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally 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. For example.

## Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle.

## Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four.

## Gordian Knot

The Gordian Knot is a legend of Phrygian Gordium associated with Alexander the Great.

## HOMFLY polynomial

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.

## Iconography

Iconography, as a branch of art history, studies the identification, description, and the 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.

## Kauffman polynomial

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

## 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.

## Loop (topology)

A loop in mathematics, in a topological space X is a continuous function f from the unit interval I.

## M. C. Escher

Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically-inspired woodcuts, lithographs, and mezzotints.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## 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.

## 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

In Norse mythology, Mjölnir (Mjǫllnir) is the hammer of Thor, the Norse god associated with thunder.

## 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, 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 either a Euclidean plane, an affine plane or a projective plane.

## 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.

In the mathematical theory of knots, a pretzel link is a special kind of link.

## Prime knot

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

## 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 "nice" decomposition as a disjoint union of circles.

## Semicubical parabola

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined by an equation of the form.

## Signature of a knot

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

## Slice knot

A slice knot is a type of mathematical knot.

## 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

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis 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.

## 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.

## Triquetra

Triquetra (Latin tri- "three" and quetrus "cornered") originally meant "triangle" and was used to refer to various three-cornered shapes.

## Unknot

The unknot arises in the mathematical theory of knots.

## Valknut

The valknut (coined from Old Norse valr, "slain warriors" and knut, "knot") is a symbol consisting of three interlocked triangles.

## Visual arts

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

## 3-sphere

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

## References

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