Trefoil knot

In knot theory

Knot theory

In mathematics, knot theory is the area of topology that studies 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 to prevent it from becoming undone. In precise mathematical language, a...

, the trefoil knot is the simplest 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...

. It can be obtained by joining the loose ends of an 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, 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 intended to be...

. It can be described as a (2,3)-torus knot

Torus knot

In knot theory, a torus knot is a special kind of knot which 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...

http://www.wolframalpha.com/input/?i=(2,3)-torus+knot, and is the closure of the 2-stranded braid

Braid group

In mathematics, the braid group on n strands, denoted by B_n, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group S_n. Here, n is a natural number; if n > 1, then B_n is an infinite group...

 σ₁³. It is also the intersection 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...

  in C² with the complex plane curve (a cuspidal cubic) of zeroes of the complex polynomial

Polynomial

In mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...

 .

The right and left-handed trefoils are the unique prime knots

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...

 which have 3-crossing diagrams

Crossing 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....

.

In knot theory

Knot theory

In mathematics, knot theory is the area of topology that studies 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 to prevent it from becoming undone. In precise mathematical language, a...

, the trefoil knot is the simplest 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...

. It can be obtained by joining the loose ends of an 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, 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 intended to be...

. It can be described as a (2,3)-torus knot

Torus knot

In knot theory, a torus knot is a special kind of knot which 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...

http://www.wolframalpha.com/input/?i=(2,3)-torus+knot, and is the closure of the 2-stranded braid

Braid group

In mathematics, the braid group on n strands, denoted by B_n, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group S_n. Here, n is a natural number; if n > 1, then B_n is an infinite group...

 σ₁³. It is also the intersection 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...

  in C² with the complex plane curve (a cuspidal cubic) of zeroes of the complex polynomial

Polynomial

In mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...

 .

Properties

The right and left-handed trefoils are the unique prime knots

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...

 which have 3-crossing diagrams

Crossing 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....

. They are chiral knots, meaning that the right-handed trefoil is the mirror image

Mirror image

A mirror image is a reflected duplication that appears identical but in reverse. As an optical effect it results from reflection off of substances such as a mirror or water...

 of the left-hand trefoil, but they are not themselves isotopic.

The simplest proof that the trefoil is not the unknot

Unknot

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...

 is that trefoil is tricolorable

Tricolorability

In the mathematical field of knot theory, the tricolorability of a knot refers to 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...

 while the unknot is not tricolorable.

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

Slice knot

A slice knot is a type of mathematical knot. It helps to remember that in knot theory, a "knot" means an embedded circle in the 3-sphereand that the 3-sphere can be thought of as the boundary of the four-dimensional ball...

, 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

Slice knot

A slice knot is a type of mathematical knot. It helps to remember that in knot theory, a "knot" means an embedded circle in the 3-sphereand that the 3-sphere can be thought of as the boundary of the four-dimensional ball...

.

The trefoil is a fibered knot

Fibered knot

A knot or link in the 3-dimensional sphere is called fibered or fibred in case there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to...

, meaning that its complement

Knot complement

In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. This solid torus is a thickened neighborhood of K. Note that the knot complement is a compact 3-manifold with boundary homeomorphic to a torus...

 in is a fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B × F, 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 which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....

 . In the model of the trefoil as the set of pairs of complex number

Complex number

A complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i² = −1...

s such that and , this fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B × F, but globally may have a different topological structure...

 has the Milnor map

Milnor map

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 encountered in the...

  as its fibration

Fibration

In mathematics, especially algebraic topology, a fibration is a surjective continuous mappingsatisfying the homotopy lifting property with respect to any space. Fiber bundles constitute important examples. In homotopy theory any mapping is 'as good as' a fibration — i.e...

, 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 and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid...

 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 -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

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 encountered in the...

.

Invariants

Its Alexander polynomial

Alexander 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...

 is and its Jones polynomial

Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1983. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer...

 is . Its 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,...

 is isomorphic to B₃, the braid group

Braid group

In mathematics, the braid group on n strands, denoted by B_n, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group S_n. Here, n is a natural number; if n > 1, then B_n is an infinite group...

 on 3 strands, which has presentation

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators...

  or

See also

• 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 symbol
  Triquetra
  Triquetra is a word derived from the Latin tri- and quetrus . Its original meaning was simply "triangle" and it has been used to refer to various three-cornered shapes. Nowadays, it has come to refer exclusively to a certain more complicated shape formed of three vesicae piscis, sometimes with an...
  
  
• Cinquefoil knot
  Cinquefoil knot
  The cinquefoil knot, also known as Solomon's seal knot, and 5₁ in most tables, is a -torus knot with five crossings. Its writhe is 5, it is invertible, but not amphichiral. It has the same Alexander polynomial as knot 10₁₃₂...