
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining 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.
Trefoil | |
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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 |
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 equations:
The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus :
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).
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.
Symmetry
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 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.
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 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
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 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 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
. The trefoil K may be viewed as the set of pairs
of complex numbers such that
and
. Then this fiber bundle has the Milnor map
as the fibre bundle projection of the knot complement
to the circle
. The fibre is a once-punctured torus. Since the knot complement is also a 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 Nε(K), and that the interior of this solid torus has been removed to create a compact knot complement
.)
Invariants
The Alexander polynomial of the trefoil knot is and the Conway polynomial is
The Jones polynomial is
and the Kauffman polynomial of the trefoil is
The HOMFLY polynomial of the trefoil is
The knot group of the trefoil is given by the presentation
or equivalently
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 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.
- 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
- A Celtic cross with trefoil knots
- A Carolingian cross
- Trefoil knot used in ATV's logo
- Mathematical surface in which the boundary is the trefoil knot in different angles
In modern art, the woodcut Knots by M. C. Escher depicts three trefoil knots whose solid forms are twisted in different ways.
See also
- Pretzel link
- Figure-eight knot (mathematics)
- Triquetra symbol
- Cinquefoil knot
- Gordian Knot
References
- Shaw, George Russell (MCMXXXIII). Knots: Useful & Ornamental, p.11. ISBN 978-0-517-46000-9.
- "3_1", The Knot Atlas.
- Weisstein, Eric W. "Trefoil Knot". MathWorld. Accessed: May 5, 2013.
- The Official M.C. Escher Website — Gallery — "Knots"
External links
- Wolframalpha: (2,3)-torus knot
- Trefoil knot 3d model
In knot theory a branch of mathematics the trefoil knot is the simplest example of a nontrivial knot The trefoil can be obtained by joining 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 TrefoilCommon nameOverhand knotArf invariant1Braid length3Braid no 2Bridge no 2Crosscap no 1Crossing no 3Genus1Hyperbolic volume0Stick no 6Tunnel no 1Unknotting no 1Conway notation 3 A B notation31Dowker notation4 6 2Last Next01 41Otheralternating torus fibered pretzel prime knot slice reversible tricolorable twist The trefoil knot is named after the three leaf clover or trefoil plant DescriptionsThe trefoil knot can be defined as the curve obtained from the following parametric equations x sin t 2sin 2ty cos t 2cos 2tz sin 3t displaystyle begin aligned x amp sin t 2 sin 2t y amp cos t 2 cos 2t z amp sin 3t end aligned The 2 3 torus knot is also a trefoil knot The following parametric equations give a 2 3 torus knot lying on torus r 2 2 z2 1 displaystyle r 2 2 z 2 1 x 2 cos 3t cos 2ty 2 cos 3t sin 2tz sin 3t displaystyle begin aligned x amp 2 cos 3t cos 2t y amp 2 cos 3t sin 2t z amp sin 3t end aligned source source source source source source source Video on making a trefoil knotOverhand knot becomes a trefoil knot by joining the ends A realization of the trefoil knot 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 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 SymmetryThe 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 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 The trefoil knot is tricolorable Form of trefoil knot without visual three fold symmetryNontrivialityThe 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 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 ClassificationIn 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 is 3 The trefoil can be described as the 2 3 torus knot It is also the knot obtained by closing the braid s13 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 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 displaystyle S 3 is a fiber bundle over the circle S1 displaystyle S 1 The trefoil K may be viewed as the set of pairs z w displaystyle z w of complex numbers such that z 2 w 2 1 displaystyle z 2 w 2 1 and z2 w3 0 displaystyle z 2 w 3 0 Then this fiber bundle has the Milnor map ϕ z w z2 w3 z2 w3 displaystyle phi z w z 2 w 3 z 2 w 3 as the fibre bundle projection of the knot complement S3 K displaystyle S 3 setminus mathbf K to the circle S1 displaystyle S 1 The fibre is a once punctured torus Since the knot complement is also a 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 Ne K and that the interior of this solid torus has been removed to create a compact knot complement S3 int Ne K displaystyle S 3 setminus operatorname int mathrm N varepsilon mathbf K InvariantsThe Alexander polynomial of the trefoil knot is D t t 1 t 1 displaystyle Delta t t 1 t 1 and the Conway polynomial is z z2 1 displaystyle nabla z z 2 1 The Jones polynomial is V q q 1 q 3 q 4 displaystyle V q q 1 q 3 q 4 and the Kauffman polynomial of the trefoil is L a z za5 z2a4 a4 za3 z2a2 2a2 displaystyle L a z za 5 z 2 a 4 a 4 za 3 z 2 a 2 2a 2 The HOMFLY polynomial of the trefoil is L a z a4 a2z2 2a2 displaystyle L alpha z alpha 4 alpha 2 z 2 2 alpha 2 The knot group of the trefoil is given by the presentation x y x2 y3 displaystyle langle x y mid x 2 y 3 rangle or equivalently x y xyx yxy displaystyle langle x y mid xyx yxy rangle This group is isomorphic to the braid group with three strands In religion and cultureAs 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 An ancient Norse Mjollnir pendant with trefoils A simple triquetra symbol A tightly knotted triquetra The Germanic Valknut A metallic Valknut in the shape of a trefoil A Celtic cross with trefoil knots A Carolingian cross Trefoil knot used in ATV s logo Mathematical surface in which the boundary is the trefoil knot in different angles In modern art the woodcut Knots by M C Escher depicts three trefoil knots whose solid forms are twisted in different ways See alsoWikimedia Commons has media related to Trefoil knots Pretzel link Figure eight knot mathematics Triquetra symbol Cinquefoil knot Gordian KnotReferencesShaw George Russell MCMXXXIII Knots Useful amp Ornamental p 11 ISBN 978 0 517 46000 9 3 1 The Knot Atlas Weisstein Eric W Trefoil Knot MathWorld Accessed May 5 2013 The Official M C Escher Website Gallery Knots External linksWolframalpha 2 3 torus knot Trefoil knot 3d model