# Trefoil knot

Trefoil Common nameOverhand knot
Arf invariant1
Braid length3
Braid no.2
Bridge no.2
Crosscap no.1
Crossing no.3
Genus1
Hyperbolic volume0
Stick no.6
Tunnel no.1
Unknotting no.1
Conway notation
A-B notation31
Dowker notation4, 6, 2
Last /Next0141
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

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

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

$x=(2+\cos 3t)\cos 2t$ $y=(2+\cos 3t)\sin 2t$ $z=\sin 3t$  Play media
Video on making a trefoil knot

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).

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.

## Symmetry

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.

## Nontriviality

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

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 .

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 $S^{3}$ is a fiber bundle over the circle $S^{1}$ . The trefoil K may be viewed as the set of pairs $(z,w)$ of complex numbers such that $|z|^{2}+|w|^{2}=1$ and $z^{2}+w^{3}=0$ . Then this fiber bundle has the Milnor map $\phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|$ as the fibre bundle projection of the knot complement $S^{3}$ \ K to the circle $S^{1}$ . The fibre is a once-punctured torus. Since the knot complement is also a Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map. (This assumes the knot has been thickened to become a solid torus Nε(K), and that the interior of this solid torus has been removed to create a compact knot complement $S^{3}$ \ int(Nε(K)).)

## Invariants

The Alexander polynomial of the trefoil knot is

$\Delta (t)=t-1+t^{-1},\,$ and the Conway polynomial is

$\nabla (z)=z^{2}+1.$ The Jones polynomial is

$V(q)=q^{-1}+q^{-3}-q^{-4},\,$ and the Kauffman polynomial of the trefoil is

$L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.\,$ The HOMFLY polynomial of the trefoil is

$L(\alpha ,z)=-\alpha ^{4}+\alpha ^{2}z^{2}+2\alpha ^{2}.\,$ The knot group of the trefoil is given by the presentation

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

$\langle x,y\mid xyx=yxy\rangle .\,$ This group is isomorphic to the braid group with three strands.

## In religion and culture

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.