Trefoil knot

Trefoil
Common name	Overhand knot
Arf invariant	1
Braid length	3
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	3
Genus	1
Hyperbolic volume	0
Stick no.	6
Tunnel no.	1
Unknotting no.	1
Conway notation	[3]
A-B notation	31
Dowker notation	4, 6, 2
Last /Next	01 / 41
Other
alternating, torus, fibered, pretzel, prime, not slice, reversible, tricolorable, twist

In knot theory, 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.

The trefoil knot is named after the three-leaf clover (or trefoil) plant.

Descriptions[edit]

The trefoil knot can be defined as the curve obtained from the following parametric equations:

${\displaystyle x=\sin t+2\sin 2t}$

${\displaystyle y=\cos t-2\cos 2t}$

${\displaystyle z=-\sin 3t}$

The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ${\displaystyle (r-2)^{2}+z^{2}=1}$:

${\displaystyle x=(2+\cos 3t)\cos 2t}$

${\displaystyle y=(2+\cos 3t)\sin 2t}$

${\displaystyle z=\sin 3t}$

Play media

Video on making a trefoil knot

Form of trefoil knot without visual three-fold symmetry

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).

A left-handed trefoil and a right-handed trefoil.

If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot.^[1]

Symmetry[edit]

The trefoil knot is chiral, 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 ambient isotopic.)

Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented 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.

The trefoil knot is tricolorable.

Overhand knot becomes a trefoil knot by joining the ends.

Nontriviality[edit]

The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves 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.

Classification[edit]

In knot theory, the trefoil is the first nontrivial knot, and is the only knot with crossing number 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 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 it does not bind 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 ${\displaystyle S^{3}}$ is a fiber bundle over the circle ${\displaystyle S^{1}}$. In the model of the trefoil as the set of pairs ${\displaystyle (z,w)}$ of complex numbers such that ${\displaystyle |z|^{2}+|w|^{2}=1}$ and ${\displaystyle z^{2}+w^{3}=0}$, this fiber bundle has the Milnor map ${\displaystyle \phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|}$ 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.

Invariants[edit]

The Alexander polynomial of the trefoil knot is

${\displaystyle \Delta (t)=t-1+t^{-1},\,}$

and the Conway polynomial is

${\displaystyle \nabla (z)=z^{2}+1.}$^[2]

The Jones polynomial is

${\displaystyle V(q)=q^{-1}+q^{-3}-q^{-4},\,}$

and the Kauffman polynomial of the trefoil is

${\displaystyle L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.\,}$

The HOMFLY polynomial of the trefoil is

${\displaystyle L(\alpha ,z)=-\alpha ^{4}+\alpha ^{2}z^{2}+2\alpha ^{2}.\,}$

The knot group of the trefoil is given by the presentation

${\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle \,}$

or equivalently

${\displaystyle \langle x,y\mid xyx=yxy\rangle .\,}$^[3]

This group is isomorphic to the braid group with three strands.

In religion and culture[edit]

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.^[4]

See also[edit]

Wikimedia Commons has media related to Trefoil knots.

• Pretzel link
• Figure-eight knot (mathematics)
• Triquetra symbol
• Cinquefoil knot
• Gordian Knot

References[edit]

External links[edit]

• Wolframalpha: (2,3)-torus knot