View source for Trefoil knot

{{Short description|Simplest non-trivial closed knot with three crossings}}
{{About|the topological concept|the protein fold|trefoil knot fold}}
{{Infobox knot theory
| name=              Trefoil
| practical name=    Overhand knot|Overhand
| image=             Blue Trefoil Knot Animated.gif
| caption=
| arf invariant=     1
| braid length=      3
| braid number=      2
| bridge number=     2
| crosscap number=   1
| crossing number=   3
| genus=             1
| hyperbolic volume= 0
| linking number=
| stick number=      6
| unknotting number= 1
| tunnel number= 1
| conway_notation=   [3]
| ab_notation=       3<sub>1</sub>
| dowker notation=   4, 6, 2
| thistlethwaite=
| last crossing=     0
| last order=        1
| next crossing=     4
| next order=        1
| alternating=       alternating
| class=             torus
| fibered=           fibered
| pretzel=           pretzel
| prime=             prime
| slice=             knot slice
| symmetry=          reversible
| tricolorable=      tricolorable
| twist=             twist
}}
In [[knot theory]], a branch of [[mathematics]], the '''trefoil knot''' is the simplest example of a nontrivial [[knot (mathematics)|knot]]. The trefoil can be obtained by joining the two loose ends of a common [[overhand knot]], resulting in a knotted [[loop (topology)|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 equation]]s:
:<math>\begin{align}
x &= \sin t + 2 \sin 2t \\
y &= \cos t - 2 \cos 2t \\
z &= -\sin 3t
\end{align}</math>

The (2,3)-[[torus knot]] is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on [[torus]] <math>(r-2)^2+z^2 = 1</math>:
:<math>\begin{align}
x &= (2+\cos 3t) \cos 2t \\
y &= (2+\cos 3t )\sin 2t \\
z &= \sin 3t
\end{align}</math>
[[File:Trefoil knot.webm|thumb|left|Video on making a trefoil knot]]
[[File:Example of Knots.svg|180px|thumb|Overhand knot becomes a trefoil knot by joining the ends.]]
[[File:Trefoil Knot.gif|thumb|180px|A realization of the trefoil knot figure]]

Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve [[Homotopy#Isotopy|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 '''C'''<sup>2</sup> of the unit [[3-sphere]] ''S''<sup>3</sup> with the [[complex plane curve]] of zeroes of the complex [[polynomial]] ''z''<sup>2</sup>&nbsp;+&nbsp;''w''<sup>3</sup> (a [[cuspidal cubic]]).

{{multiple image
| width             = 150
| footer            = A left-handed trefoil and a right-handed trefoil
| image1            = Trefoil knot left.svg
| alt1              = Left-handed trefoil
| image2            = TrefoilKnot 01.svg
| alt2              = 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.<ref>Shaw, George Russell ({{Roman|1933}}). ''Knots: Useful & Ornamental'', p.11. {{isbn|978-0-517-46000-9}}.</ref>

== Symmetry ==
The trefoil knot is [[chirality (mathematics)|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 isotopy|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.

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

[[File:Tricoloring.png|thumb|180px|The trefoil knot is [[tricolorability|tricolorable]].]]
[[File:Trefoil-non-3-symm.svg|thumb|right|Form of trefoil knot without visual three-fold symmetry]]
== 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 move]]s 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 (knot theory)|crossing number]] three. It is a [[prime knot]], and is listed as 3<sub>1</sub> in the [[Alexander-Briggs notation]]. The [[Dowker notation]] for the trefoil is 4&nbsp;6&nbsp;2, and the [[Conway notation (knot theory)|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 group|braid]] σ<sub>1</sub><sup>3</sup>.

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 of a knot|signature]] is not zero. Another proof is that its Alexander polynomial does not satisfy the [[slice knot|Fox-Milnor condition]].

The trefoil is a [[fibered knot]], meaning that its [[knot complement|complement]] in <math>S^3</math> is a [[fiber bundle]] over the [[circle]] <math>S^1</math>. The trefoil '''K''' may be viewed as the set of pairs <math>(z,w)</math> of [[complex number]]s such that <math>|z|^2+|w|^2=1</math> and <math>z^2+w^3=0</math>. Then this [[fiber bundle]] has the [[Milnor map]] <math>\phi(z, w) = (z^2+w^3) / |z^2+w^3|</math> as the fibre bundle projection of the knot complement <math>S^3 \setminus \mathbf{K}</math> to the circle <math>S^1</math>. The fibre is a [[Puncture (topology)|once-punctured]] [[torus]]. Since the knot complement is also a [[Seifert fiber space|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<sub>ε</sub>('''K'''), and that the interior of this solid torus has been removed to create a compact knot complement <math>S^3 \setminus \operatorname{int}(\mathrm{N}_\varepsilon(\mathbf{K})</math>.)

== Invariants ==

The [[Alexander polynomial]] of the trefoil knot is<math display="block">\Delta(t) = t - 1 + t^{-1}, </math>since <math>\begin{pmatrix}1 & -1 \\ 0 & 1\end{pmatrix}</math> is a possible [[Seifert surface|Seifert matrix]] (for the left-hand one), or because of its [[Alexander polynomial#Alexander–Conway polynomial|Conway polynomial]],<ref>{{Knot Atlas|3_1}}</ref> which is<math display="block">\nabla(z) = z^2 + 1.</math>The [[Jones polynomial]] is<math display="block">V(q) = q^{-1} + q^{-3} - q^{-4}, </math>and the [[Kauffman polynomial]] of the trefoil is<math display="block">L(a,z) = za^5 + z^2a^4 - a^4 + za^3 + z^2a^2-2a^2. </math>The [[HOMFLY polynomial]] of the trefoil is<math display="block">L(\alpha,z) = -\alpha^4 + \alpha^2z^2 + 2\alpha^2. </math>

The [[knot group]] of the trefoil is given by the presentation<math display="block">\langle x,y\mid x^2=y^3\rangle, </math>or equivalently<ref>{{MathWorld|title=Trefoil Knot|id=TrefoilKnot}} by Wolfram. Accessed: May 5, 2013.</ref><math display="block">\langle x,y \mid xyx=yxy \rangle. </math>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 (visual arts)|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]].

{{Gallery
|File:Mjollnir.png|An ancient Norse [[Mjöllnir]] pendant with trefoils
|File:Triquetra-Vesica.svg|A simple [[triquetra]] symbol
|File:Triquetra-tightly-knotted.svg|A tightly-knotted triquetra
|File:Valknut-Symbol-triquetra.svg|The Germanic [[Valknut]]
|File:Metallic Valknut black background.PNG|A metallic Valknut in the shape of a trefoil
|File:Cruz_Celta_con_Trisquetas.svg|A [[Celtic cross]] with trefoil knots
|File:Triquetra-Cross.svg|A [[Carolingian cross]]
|File:ATV News GL5463 left side 20060712.jpg|Trefoil knot used in [[ATV Home|ATV]]'s logo
|File:Superfície não orientável - Bordo trifólio.jpg|Mathematical surface in which the boundary is the trefoil knot in different angles
}}
In modern art, the woodcut ''Knots'' by [[M. C. Escher]] (1965) depicts three trefoil knots whose solid forms are twisted in different ways.<ref>[https://www.mcescher.com/gallery/recognition-success/knots/ The Official M.C. Escher Website — Gallery — "Knots"] ''www.mcescher.com'' In black/green/brown via 3 woodcuts. Catalogue nr 444. The Hague, Municipal Museum (1981).</ref>

== See also ==
{{commons category|Trefoil knots}}
* [[Pretzel link]]
* [[Figure-eight knot (mathematics)]]
* [[Triquetra]] symbol
* [[Cinquefoil knot]]
* [[Gordian Knot]]

== References ==

{{reflist | 30em}}

== External links ==
* [https://www.wolframalpha.com/input/?i=(2,3)-torus+knot Wolframalpha: (2,3)-torus knot]
* [https://www.cgtrader.com/free-3d-models/various/various-models/torus-knot Trefoil knot 3d model] ''www.cgtrader.com''

{{Knot theory|state=collapsed}}{{Category link|Topology}}