Trefoil knot
Encyclopedia
In topology
, a branch of mathematics
, the trefoil knot is the simplest example of a nontrivial knot
. The trefoil can be obtained by joining together the two loose ends of a common overhand knot
, resulting in a knotted loop
. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory
, which has diverse applications in topology
, geometry
, physics
, and chemistry
.
The trefoil knot is named after the three-leaf clover
(or trefoil) plant.
s:
This curve lies entirely on the torus
, making the trefoil the simplest example of a torus knot
. (Specifically, the trefoil is the (2,3)-torus knot, since the curve winds around the torus three times in one direction and twice in the other direction.)
Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mirror image
of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation.
In algebraic geometry
, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere
S3 with the complex plane curve of zeroes of the complex polynomial
z2 + w3 (a cuspidal cubic).
, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice-versa. (That is, the two trefoils are not isotopic.)
Though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a counterclockwise-oriented trefoil and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve.
. In particular, there is no sequence of Reidemeister move
s that will untie a trefoil.
Proving this requires the construction of a knot invariant
that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability
: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial
distinguishes the trefoil from an unknot, as do most other strong knot invariants.
three. It is a prime knot
, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation
for the trefoil is 4 6 2, and the Conway notation
for the trefoil is [3].
The trefoil can be described as the (2,3)-torus knot
. It is also the knot obtained by closing the braid
σ13.
The trefoil is an alternating knot
. However, it is not a slice knot, meaning that it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature
is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition.
The trefoil is a fibered knot
, meaning that its complement
in
is a fiber bundle
over the circle
. In the model of the trefoil as the set of pairs 
of complex number
s such that
and 
, this fiber bundle
has the Milnor map
as its fibration
, and a once-punctured torus
as its fiber surface. Since the knot complement is Seifert fibred
with boundary, it has a horizontal incompressible surface -- this is also the fiber of the Milnor map
.
of the trefoil knot is
and the Conway polynomial is
The Jones polynomial is
and the Kauffman polynomial of the trefoil is
The knot group
of the trefoil is given by the presentation
or equivalently
This group is isomorphic to the braid group
with three strands.
As the simplest nontrivial knot, the trefoil is a common motif in iconography
and the visual arts
. For example, the common form of the triquetra
symbol is a trefoil, as are some versions of the Germanic Valknut
.
In modern art, the woodcut “Knots” by M. C. Escher
depicts three trefoil knots whose solid forms are twisted in different ways.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the trefoil knot is the simplest example of a nontrivial knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
. The trefoil can be obtained by joining together the two loose ends of a common overhand knot
Overhand knot
The overhand knot is one of the most fundamental knots and forms the basis of many others including the simple noose, overhand loop, angler's loop, reef knot, fisherman's knot and water knot. The overhand knot is very secure, to the point of jamming badly. It should be used if the knot is...
, resulting in a knotted loop
Loop (topology)
In mathematics, a loop in a topological space X is a path f from the unit interval I = [0,1] to X such that f = f...
. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, which has diverse applications in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
.
The trefoil knot is named after the three-leaf clover
Clover
Clover , or trefoil, is a genus of about 300 species of plants in the leguminous pea family Fabaceae. The genus has a cosmopolitan distribution; the highest diversity is found in the temperate Northern Hemisphere, but many species also occur in South America and Africa, including at high altitudes...
(or trefoil) plant.
Descriptions
The trefoil can be defined as the curve obtained from the following parametric equationParametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
s:
This curve lies entirely on the torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
, making the trefoil the simplest example of a torus knotTorus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
. (Specifically, the trefoil is the (2,3)-torus knot, since the curve winds around the torus three times in one direction and twice in the other direction.)
Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mirror image
Mirror image
A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...
of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation.
In algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
S3 with the complex plane curve of zeroes of the complex polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
z2 + w3 (a cuspidal cubic).
Symmetry
The trefoil knot is chiralChirality (mathematics)
In geometry, a figure is chiral 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. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice-versa. (That is, the two trefoils are not isotopic.)
Though the trefoil knot is chiral, it is also invertible, meaning that there is no distinction between a counterclockwise-oriented trefoil and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve.
Nontriviality
The trefoil knot is nontrivial, meaning that it is not possible to “untie” a trefoil knot in three dimensions without cutting it. From a mathematical point of view, this means that a trefoil knot is not isotopic to the unknotUnknot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
. In particular, there is no sequence of Reidemeister move
Reidemeister move
In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. In 1926, Kurt Reidemeister and independently, in 1927, J.W. Alexander and G.B...
s that will untie a trefoil.
Proving this requires the construction of a knot invariant
Knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability
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. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different knots...
: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial
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.-History:The first knot polynomial, the Alexander polynomial, was introduced by J. W...
distinguishes the trefoil from an unknot, as do most other strong knot invariants.
Classification
In knot theory, the trefoil is the first nontrivial knot, and is the only knot with crossing numberCrossing number (knot theory)
In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
three. It is a prime knot
Prime knot
In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite. It can be a nontrivial problem to determine whether a...
, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation
Dowker notation
thumb|200px|A knot diagram with crossings labelled for a Dowker sequenceIn the mathematical field of knot theory, the Dowker notation, also called the Dowker–Thistlethwaite notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and...
for the trefoil is 4 6 2, and the Conway notation
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. It composes a knot using certain operations on tangles to construct it.-Tangles:...
for the trefoil is [3].
The trefoil can be described as the (2,3)-torus knot
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. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
. It is also the knot obtained by closing the braid
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
σ13.
The trefoil is an alternating knot
Alternating knot
In knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. A link is alternating if it has an alternating diagram....
. However, it is not a slice knot, meaning that it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature
Signature of a knot
The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K...
is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition.
The trefoil is a fibered knot
Fibered knot
A knot or link Kin the 3-dimensional sphere S^3 is called fibered or fibred 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...
, meaning that its complement
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
in
is a fiber bundleFiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
over the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. In the model of the trefoil as the set of pairs of complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s such that
and , this fiber bundleFiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
has the Milnor map
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 and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly...
as its fibrationFibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
, and a once-punctured torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
as its fiber surface. Since the knot complement is Seifert fibred
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
with boundary, it has a horizontal incompressible surface -- this is also the fiber of the Milnor map
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 and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly...
.
Invariants
The Alexander polynomialAlexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
of the trefoil knot is
and the Conway polynomial is
The Jones polynomial is
and the Kauffman polynomial of the trefoil is
The knot group
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,\pi_1....
of the trefoil is given by the presentation
or equivalently
This group is isomorphic to the braid group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
with three strands.
Trefoils in religion and culture
Iconography
Iconography is the branch of art history which studies the identification, description, and the interpretation of the content of images. The word iconography literally means "image writing", and comes from the Greek "image" and "to write". A secondary meaning is the painting of icons in the...
and the visual arts
Visual arts
The visual arts are art forms that create works which are primarily visual in nature, such as ceramics, drawing, painting, sculpture, printmaking, design, crafts, and often modern visual arts and architecture...
. For example, the common form of the triquetra
Triquetra
Triquetra originally meant "triangle" and was used to refer to various three-cornered shapes. Nowadays, it has come to refer exclusively to a particular more complicated shape formed of three vesicae piscis, sometimes with an added circle in or around it...
symbol is a trefoil, as are some versions of the Germanic Valknut
Valknut
The Valknut is a symbol consisting of three interlocked triangles, and appears on various Germanic objects. A number of theories have been proposed for its significance....
.
In modern art, the woodcut “Knots” by M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
depicts three trefoil knots whose solid forms are twisted in different ways.
See also
- Pretzel knot
- Figure-eight knot (mathematics)Figure-eight knot (mathematics)In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
- Triquetra symbolTriquetraTriquetra originally meant "triangle" and was used to refer to various three-cornered shapes. Nowadays, it has come to refer exclusively to a particular more complicated shape formed of three vesicae piscis, sometimes with an added circle in or around it...
- Cinquefoil knotCinquefoil knotIn 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. It is listed as the 51 knot in the Alexander-Briggs notation, and can also be described as the -torus knot...