Trefoil knot

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This article is about the topological concept. For the protein fold, see trefoil knot fold.
Trefoil
Blue Trefoil Knot.png
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 0141
Other
alternating, torus, fibered, pretzel, prime, slice, reversible, tricolorable, twist

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:

The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus :

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 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 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 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 is a fiber bundle over the circle . In the model of the trefoil as the set of pairs of complex numbers such that and , this fiber bundle has the Milnor map 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

and the Conway polynomial is

[2]

The Jones polynomial is

and the Kauffman polynomial of the trefoil is

The knot group of the trefoil is given by the presentation

or equivalently

[3]

This group is isomorphic to the braid group with three strands.

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

External links[edit]