In
knot theory, a branch of
mathematics, the trefoil knot is the simplest example of a nontrivial
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
. The trefoil can be obtained by joining together the two loose ends of a common
overhand knot
The overhand knot is one of the most fundamental knots, and it forms the basis of many others, including the simple noose, overhand loop, angler's loop, reef knot, fisherman's knot, Half hitch, and water knot. The overhand knot is a stopper, ...
, 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
Clover or trefoil are common names for plants of the genus ''Trifolium'' (from Latin ''tres'' 'three' + ''folium'' 'leaf'), consisting of about 300 species of flowering plants in the legume or pea family Fabaceae originating in Europe. The genus ...
(or trefoil) plant.
Descriptions
The trefoil knot can be defined as the
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
obtained from the following
parametric equations:
:
The (2,3)-
torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...
is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
:
:
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 C
2 of the unit
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
''S''
3 with the
complex plane curve of zeroes of the complex
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
''z''
2 + ''w''
3 (a
cuspidal cubic
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
: y^2 - a^2 x^3 = 0
(with ) in some Cartesian coordinate system.
Solving for leads to the ''explicit form''
: y = \ ...
).
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
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from i ...
, 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
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
-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.
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 the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
. In particular, there is no sequence of
Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...
s that will untie a trefoil.
Proving this requires the construction of a
knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...
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
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
, and is listed as 3
1 in the
Alexander-Briggs notation. The
Dowker notation for the trefoil is 4 6 2, and the
Conway notation is
The trefoil can be described as the (2,3)-
torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...
. It is also the knot obtained by closing the
braid σ
13.
The trefoil is an
alternating knot
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.
Many of the knots with crossing ...
. However, it is not a
slice knot
A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space.
Definition
A knot K \subset S^3 is said to be a topologically or smoothly slice knot, if it is the boundary of an embedded disk in ...
, 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
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
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
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
in
is a
fiber bundle over the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
. The trefoil K may be viewed as the set of pairs
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s such that
and
. Then this
fiber bundle has the
Milnor map In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied ...
as the fibre bundle projection of the knot complement
to the circle
. The fibre is a
once-punctured torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
. 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 In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied ...
. (This assumes the knot has been thickened to become a solid torus N
ε(K), and that the interior of this solid torus has been removed to create a compact knot complement
.)
Invariants
The
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
of the trefoil knot is
and the
Conway polynomial is
The
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomi ...
is
and the
Kauffman polynomial
In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as
:F(K)(a,z)=a^L(K)\,,
where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...
of the trefoil is
The
HOMFLY polynomial of the trefoil is
The
knot group
In mathematics, a knot (mathematics), knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3,
:\pi_1(\mathbb^3 \setminus K).
Oth ...
of the trefoil is given by the presentation
or equivalently
This group is isomorphic to the
braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
with three strands.
In religion and culture
As the simplest nontrivial knot, the trefoil is a common
motif in
iconography and the
visual arts
The visual arts are art forms such as painting, drawing, printmaking, sculpture, ceramics, photography, video, filmmaking, design, crafts and architecture. Many artistic disciplines such as performing arts, conceptual art, and textile art ...
. For example, the common form of the
triquetra symbol is a trefoil, as are some versions of the Germanic
Valknut
The valknut is a symbol consisting of three interlocked triangles. It appears on a variety of objects from the archaeological record of the ancient Germanic peoples. The term ''valknut'' is a modern development; it is not known what term or term ...
.
In modern art, the woodcut ''Knots'' by
M. C. Escher depicts three trefoil knots whose solid forms are twisted in different ways.
The Official M.C. Escher Website — Gallery — "Knots"
/ref>
See also
* Pretzel link
In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number tangles made of two intertwined circular helices. The tangles are connected cyclicly, the first component of the first tangle is con ...
* Figure-eight knot (mathematics)
Figure 8 (figure of 8 in British English) may refer to:
* 8 (number), in Arabic numerals
Entertainment
* ''Figure 8'' (album), a 2000 album by Elliott Smith
* "Figure of Eight" (song), a 1989 song by Paul McCartney
* ''Figure Eight EP'', a ...
* Triquetra symbol
* Cinquefoil knot
* Gordian Knot
References
External links
Wolframalpha: (2,3)-torus knot
Trefoil knot 3d model
{{Knot theory, state=collapsed