Sunday, February 26, 2017

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Trefoil knot

From Wikipedia, the free encyclopedia
This article is about the topological concept. For the protein fold, see trefoil knot fold.
Trefoil
Blue Trefoil Knot.png
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[3]
A-B notation31
Dowker notation4, 6, 2
Last /Next01 / 41
Other
alternatingtorusfiberedpretzelprimeslicereversibletricolorabletwist
In topology, 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[edit]

The trefoil knot can be defined as the curve obtained from the following parametric equations:
xsint2sin2t
ycost2cos2t
zsin3t
The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus r22z21:
x2cos3tcos2t
y2cos3tsin2t
zsin3t
File:Trefoil knot.webm
Video on making a trefoil knot
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 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).
Left-handed trefoil
Right-handed trefoil
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 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.
The trefoil knot is tricolorable.
Overhand knot becomes a trefoil knot by joining the ends.

Nontriviality[edit]

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 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 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 Ïƒ13.
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 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 S3 is a fiber bundle over the circle S1. In the model of the trefoil as the set of pairs zw of complex numbers such that z2w21 and z2w30, this fiber bundle has the Milnor map Ï•zwz2w3z2w3 as its fibration, and a once-punctured torus as its fiber surface. Since the knot complement is Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map.

Invariants[edit]

Trefoil Knot.gif
The Alexander polynomial of the trefoil knot is
Δtt1t1
and the Conway polynomial is
zz21.[2]
Vqq1q3q4
and the Kauffman polynomial of the trefoil is
Lazza5z2a4a4za3z2a22a2
The knot group of the trefoil is given by the presentation
xyx2y3
or equivalently
xyxyxyxy[3]
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.
Trefoil knots
An ancient Norse Mjöllnirpendant with trefoils 
A simple triquetra symbol 
A tightly-knotted triquetra 
The Germanic Valknut 
A metallic Valknut in the shape of a trefoil 
Trefoil knot used in 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.[4]

See also[edit]

References[edit]

  1. Jump up ^ Shaw, George Russell (MCMXXXIII). Knots: Useful & Ornamental, p.11. ISBN 978-0-517-46000-9.
  2. Jump up ^ "3_1", The Knot Atlas.
  3. Jump up ^ Weisstein, Eric W. "Trefoil Knot"MathWorld. Accessed: May 5, 2013.
  4. Jump up ^ The Official M.C. Escher Website — Gallery — "Knots"

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