In knot theory, the trefoil knot is the simplest nontrivial knot. Diagram of a right trefoil knot. ... Image File history File links Size of this preview: 800 × 600 pixelsFull resolution (1024 × 768 pixel, file size: 100 KB, MIME type: image/png) Description: Trefoil Knot Source: self-made Date: created 24 Jun. ... Trefoil knot, the simplest non-trivial knot. ... A trefoil knot. ...

Descriptions

• It can be obtained by joining the loose ends of an overhand knot.
• It can be described as a (2,3)-torus knot.
• It is the closure of the braid σ₁³.
• It is the intersection of the unit 3-sphere S³ in C² with the complex plane curve (a cuspidal cubic) of zeroes of the complex polynomial z² + w³.

The overhand knot is a type of knot. ... A (3,7)-3D torus knot rendered by Apple Grapher. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

Properties

• It is the unique prime knot with three crossings.
• It is chiral, meaning it is not equivalent to its mirror image.
• It is alternating.
• 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.
• It 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 (z,w) of complex numbers such that | z | ² + | w | ² = 1 and z² + w³ = 0, this fiber bundle has the Milnor map φ(z,w) = ( < gallery > z⁴ + w³ < / gallery > ) / | z² + w³ | as its fibration, and a once-punctured torus as its fiber surface.

In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. ... In mathematics, crossing numbers arise in two related contexts: in knot theory and in graph theory. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... A mirror image is a mirror based duplicate of a single image. ... One of three non-alternating knots with crossing number 8 In knot theory, a knot diagram is alternating if the crossings alternate under, over, under, over, as you travel along the strand. ... A slice knot is a type of mathematical knot. ... In mathematics, the signature of a knot K, denoted by s(K), can be defined using the skein relationship and where is the knot polynomial and is an odd number. ... A knot or link in the 3-dimensional sphere is called fibered (sometimes spelled fibred) in case there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to then the intersection of and... In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... 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. ... This article may be too technical for most readers to understand. ... In geometry, a torus (pl. ...

Invariants

• Its Alexander polynomial is t² − t + 1.
• Its Jones polynomial is t + t³ − t⁴.
• Its knot group is isomorphic to B₃, the braid group on 3 strands, which has presentation or

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ... This article needs cleanup. ... In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the braid group on n strands, denoted by Bn, is a certain group which has a nice geometrical representation and in a sense generalizes the symmetric group Sn. ... In mathematics, one method of defining a group is by a presentation. ...

See also

• Figure-eight knot (mathematics)
• Triquetra symbol

this is a really gay knot, it doesnt even have an end or start for the rope!!! lol In knot theory, a figure-eight knot is the unique knot with a crossing number of four, the smallest possible except for the unknot and trefoil knot. ... Close-up of a triquetra on one of the Funbo Runestones. ...