Trefoil knot – Wikipedia

Posted on March 27, 2015 by lordneo

Simplest non-trivial closed knot with three crossings

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
alternating, torus, fibered, pretzel, prime, knot 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.

Table of Contents

Descriptions[edit]

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

{displaystyle {begin{aligned}x&=sin t+2sin 2t\y&=cos t-2cos 2t\z&=-sin 3tend{aligned}}}

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}

$(r-2)^{2}+z^{2}=1$ :

{displaystyle {begin{aligned}x&=(2+cos 3t)cos 2t\y&=(2+cos 3t)sin 2t\z&=sin 3tend{aligned}}}

Video on making a trefoil knot

Overhand knot becomes a trefoil knot by joining the ends.

A realization of the 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).

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.

Form of trefoil knot without visual three-fold symmetry

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 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

{displaystyle S^{3}}

$S^{3}$ is a fiber bundle over the circle

{displaystyle S^{1}}

$S^{1}$ . The trefoil K may be viewed as the set of pairs

{displaystyle (z,w)}

$(z,w)$ of complex numbers such that

{displaystyle |z|^{2}+|w|^{2}=1}

$|z|^{2}+|w|^{2}=1$ and

{displaystyle z^{2}+w^{3}=0}

$z^{2}+w^{3}=0$ . Then this fiber bundle has the Milnor map

{displaystyle phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|}

${displaystyle phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|}$ as the fibre bundle projection of the knot complement

{displaystyle S^{3}setminus mathbf {K} }

${displaystyle S^{3}setminus mathbf {K} }$ to the circle

{displaystyle S^{1}}

$S^{1}$ . 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 N_ε(K), and that the interior of this solid torus has been removed to create a compact knot complement

{displaystyle S^{3}setminus operatorname {int} (mathrm {N} _{varepsilon }(mathbf {K} )}

${displaystyle S^{3}setminus operatorname {int} (mathrm {N} _{varepsilon }(mathbf {K} )}$ .)

Invariants[edit]

The Alexander polynomial of the trefoil knot is

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

and the Conway polynomial is^[2]

{displaystyle nabla (z)=z^{2}+1.}

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}+2alpha ^{2}.}

The knot group of the trefoil is given by the presentation

{displaystyle langle x,ymid x^{2}=y^{3}rangle }

or equivalently^[3]

{displaystyle langle x,ymid xyx=yxyrangle .}

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]

Trefoil knot – Wikipedia

Descriptions[edit]

Symmetry[edit]

Nontriviality[edit]

Classification[edit]

Invariants[edit]

In religion and culture[edit]

See also[edit]

References[edit]

External links[edit]

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