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, knot slice, reversible, tricolorable, twist

Inknot theory, a branch ofmathematics, thetrefoil knot is the simplest example of a nontrivialknot. The trefoil can be obtained by joining the two loose ends of a commonoverhand knot, resulting in a knottedloop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.

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

The trefoil knot can be defined as thecurve obtained from the followingparametric equations:

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

Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curveisotopic to a trefoil knot is also considered to be a trefoil. In addition, themirror image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using aknot diagram instead of an explicit parametric equation.

Inalgebraic geometry, the trefoil can also be obtained as the intersection inC² of the unit3-sphereS³ with thecomplex plane curve of zeroes of the complexpolynomialz² + w³ (acuspidal 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]

The trefoil knot ischiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as theleft-handed trefoil and theright-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 notambient isotopic.)

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

But the knot has rotational symmetry. The axis is about a line perpendicular to the page for the 3-coloured image.

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 theunknot. In particular, there is no sequence ofReidemeister moves that will untie a trefoil.

Proving this requires the construction of aknot invariant that distinguishes the trefoil from the unknot. The simplest such invariant istricolorability: the trefoil is tricolorable, but the unknot is not. In addition, virtually every majorknot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants.

In knot theory, the trefoil is the first nontrivial knot, and is the only knot withcrossing number three. It is aprime knot, and is listed as 3₁ in theAlexander-Briggs notation. TheDowker notation for the trefoil is 4 6 2, and theConway notation is [3].

The trefoil can be described as the (2,3)-torus knot. It is also the knot obtained by closing thebraid σ₁³.

The trefoil is analternating knot. However, it is not aslice 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 itssignature is not zero. Another proof is that its Alexander polynomial does not satisfy theFox-Milnor condition.

The trefoil is afibered knot, meaning that itscomplement in${\displaystyle S^3}$ is afiber bundle over thecircle${\displaystyle S^1}$. The trefoilK may be viewed as the set of pairs${\displaystyle (z,w)}$ ofcomplex numbers such that${\displaystyle |z{|}^2+|w{|}^2=1}$ and${\displaystyle z^2+w^3=0}$. Then thisfiber bundle has theMilnor map${\displaystyle \varphi (z,w)=(z^2+w^3)/|z^2+w^3|}$ as the fibre bundle projection of the knot complement${\displaystyle S^3\setminus \mathbf{K}}$ to the circle${\displaystyle S^1}$. The fibre is aonce-puncturedtorus. Since the knot complement is also aSeifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of theMilnor 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${\displaystyle S^3\setminus \mathrm{int}({\mathrm{N}}_{\epsilon }(\mathbf{K})}$.)

TheAlexander polynomial of the trefoil knot is$${\displaystyle \mathrm{\Delta }(t)=t-1+t^{-1},}$$and theConway polynomial is^[2]$${\displaystyle \mathrm{\nabla }(z)=z^2+1.}$$TheJones polynomial is$${\displaystyle V(q)=q^{-1}+q^{-3}-q^{-4},}$$and theKauffman polynomial of the trefoil is$${\displaystyle L(a,z)=z{a}^5+z^2{a}^4-a^4+z{a}^3+z^2{a}^2-2a^2.}$$TheHOMFLY polynomial of the trefoil is$${\displaystyle L(\alpha ,z)=-{\alpha }^4+{\alpha }^2{z}^2+2{\alpha }^2.}$$Theknot group of the trefoil is given by the presentation$${\displaystyle ⟨x,y\mid x^2=y^3⟩}$$or equivalently^[3]$${\displaystyle ⟨x,y\mid xyx=yxy⟩.}$$This group is isomorphic to thebraid group with three strands.

As the simplest nontrivial knot, the trefoil is a commonmotif iniconography and thevisual arts. For example, the common form of thetriquetra symbol is a trefoil, as are some versions of the GermanicValknut.

In modern art, the woodcutKnots byM. C. Escher depicts three trefoil knots whose solid forms are twisted in different ways.^[4]

Wikimedia Commons has media related toTrefoil knots.

• Pretzel link
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• Wolframalpha: (2,3)-torus knot
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• Knot theory
• Alternating knots and links
• Torus knots and links
• Fibered knots and links
• Pretzel knots and links (mathematics)
• Prime knots and links
• Slice knots and links
• Reversible knots and links
• Tricolorable knots and links
• Twist knots

• Articles with short description
• Short description matches Wikidata
• Commons category link is on Wikidata