en>Arthur Rubin: Reverted good faith edits by VladimirReshetnikov (talk): I don't know what the big union character is, but the little one is confusing. (TW)

2012-07-02T04:58:08Z

Reverted good faith edits by VladimirReshetnikov (talk): I don't know what the big union character is, but the little one is confusing. (TW)

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{{Otheruses4|the topological concept|the protein fold|trefoil knot fold}}

{{Infobox knot theory
| name= Trefoil
| practical name= Overhand knot|Overhand
| image= Blue Trefoil Knot.png
| caption=
| arf invariant= 1
| braid length= 3
| braid number= 2
| bridge number= 2
| crosscap number= 1
| crossing number= 3
| hyperbolic volume= 0
| linking number=
| stick number= 6
| unknotting number= 1
| conway_notation= [3]
| ab_notation= 31
| 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= slice
| symmetry= reversible
| tricolorable= tricolorable
| twist= twist
}}

In [[topology]], a branch of [[mathematics]], the '''trefoil knot''' is the simplest example of a nontrivial [[knot (mathematics)|knot]]. The trefoil can be obtained by joining together 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]], which has diverse applications in [[topology]], [[geometry]], [[physics]], [[chemistry]] and [[Magic (illusion)|magic]].

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>x = \sin t + 2 \sin 2t</math>
:<math>\qquad y=\cos t - 2 \cos 2t</math>
:<math>\qquad z=-\sin 3t</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>x = (2+\cos 3t)\cos 2t</math>
:<math>\qquad y=(2+\cos 3t )\sin 2t</math>
:<math>\qquad z=\sin 3t</math>

[[File:Trefoil-non-3-symm.svg|thumb|right|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 [[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'''2 of the unit [[3-sphere]] ''S''3 with the [[complex plane curve]] of zeroes of the complex [[polynomial]] ''z''2 + ''w''3 (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, a trefoil knot results.<ref>Shaw, George Russell ({{Roman|1933}}). ''Knots: Useful & Ornamental'', p.11. {{pre ISBN}}.</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 isotopic.)

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

[[Image:Tricoloring.png|thumb|180px|The trefoil knot is [[tricolorability|tricolorable]].]]
[[Image:Example of Knots.svg|180px|thumb|Overhand knot becomes a trefoil knot by joining the ends.]]

==Nontriviality==
The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. From a mathematical point of view, 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 31 in the [[Alexander-Briggs notation]]. The [[Dowker notation]] for the trefoil is 4 6 2, and the [[Conway notation (knot theory)|Conway notation]] for the trefoil 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]] σ13.

The trefoil is an [[alternating knot]]. However, it is not a [[slice knot]], meaning that 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>. In the model of the trefoil 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>, this [[fiber bundle]] has the [[Milnor map]] <math>\phi(z,w)=(
z^2+w^3)/|z^2+w^3|</math> as its [[fibration]], and a once-punctured [[torus]] as its [[fiber surface]]. Since the knot complement is [[Seifert fiber space|Seifert fibred]] with boundary, it has a horizontal incompressible surface -- this is also the fiber of the [[Milnor map]].

==Invariants==
The [[Alexander polynomial]] of the trefoil knot is
:<math>\Delta(t) = t - 1 + t^{-1}, \, </math>
and the [[Alexander polynomial#Alexander.E2.80.93Conway polynomial|Conway polynomial]] is
:<math>\nabla(z) = z^2 + 1.</math><ref>{{Knot Atlas|3_1}}</ref>
The [[Jones polynomial]] is
:<math>V(q) = q^{-1} + q^{-3} - q^{-4}, \, </math>
and the [[Kauffman polynomial]] of the trefoil is
:<math>L(a,z) = za^5 + z^2a^4 - a^4 + za^3 + z^2a^2-2a^2. \, </math>
The [[knot group]] of the trefoil is given by the presentation
:<math>\langle x,y \mid x^2=y^3 \rangle \, </math>
or equivalently
:<math>\langle x,y \mid xyx=yxy \rangle. \, </math><ref>{{MathWorld|title=Trefoil Knot|id=TrefoilKnot}} Accessed: May 5, 2013.</ref>
This group is isomorphic to the [[braid group]] with three strands.

==Trefoils 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
|title=Trefoil knots
|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:ATV NewsCar.jpg|Trefoil knot used in [[ATV Home|aTV]]'s logo
}}
In modern art, the woodcut ''Knots'' by [[M. C. Escher]] depicts three trefoil knots whose solid forms are twisted in different ways.<ref>[http://www.mcescher.com/Gallery/recogn-bmp/LW444.jpg The Official M.C. Escher Website — Gallery — "Knots"]</ref>

==See also==
*[[Pretzel link]]
*[[Figure-eight knot (mathematics)]]
*[[Triquetra|Triquetra symbol]]
*[[Cinquefoil knot]]
*[[Gordian Knot]]

==References==
{{reflist}}

==External links==
*[http://www.wolframalpha.com/input/?i=(2,3)-torus+knot Wolframalpha: (2,3)-torus knot]

{{Knot theory|state=collapsed}}

Normal function - Revision history

en>Arthur Rubin: Reverted good faith edits by VladimirReshetnikov (talk): I don't know what the big union character is, but the little one is confusing. (TW)