Knot (mathematics) - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a knot is an embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

of a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

in 3-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, R³, considered up to continuous deformations (isotopies

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

). A crucial difference between the standard mathematical and conventional notions of a knot

Knot

A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object—the "load"...

is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of

, especially in the case

. The branch of mathematics that studies knots is known as knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

Types of knots

The simplest knot, called the unknot
Unknot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...

, is a round circle embedded in R³

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot

Trefoil knot

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...

(3₁ in the table), the figure-eight knot

Figure-eight knot (mathematics)

In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...

(4₁) and the cinquefoil knot

Cinquefoil knot

In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, and can also be described as the -torus knot...

(5₁).

Several knots, possibly tangled together, are called links

Link (knot theory)

In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...

. Knots are links with a single component.

Often mathematicians prefer to consider knots embedded into the 3-sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

, S³, rather than R³ since the 3-sphere is compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

. The 3-sphere is equivalent to R³ with a single point added at infinity (see one-point compactification).

A knot is tame if it can be "thickened up", that is, if there exists an extension to an embedding of the solid torus

Solid torus

In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...

, into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain

Polygonal chain

A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points \scriptstyle called its vertices so that the curve consists of the line segments connecting the...

. Knots that are not tame are called wild and can have pathological

Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....

behavior. In knot theory and 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.

Given a knot in the 3-sphere, the knot complement

Knot complement

In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...

is all the points of the 3-sphere not contained in the knot. A major theorem of Gordon and Luecke states that at most two knots have homeomorphic complements (the original knot and its mirror reflection). This in effect turns the study of knots into the study of their complements, and in turn into 3-manifold theory.

The JSJ decomposition

JSJ decomposition

In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...

and Thurston's hyperbolization theorem

Geometrization conjecture

Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...

reduces the study of knots in the 3-sphere to the study of various geometric manifolds via splicing or satellite operations.

Satellite knot

In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non-boundary parallel torus in its complement. The class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite knot K can be picturesquely described as follows:...

In the pictured knot, the JSJ-decomposition splits the complement into the union of three manifolds: two trefoil complements

Trefoil knot

and the complement of the Borromean rings

Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...

. The trefoil complement has the geometry of

, while the Borromean rings complement has the geometry of

Knots, more generally speaking

In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold

with a submanifold

, one sometimes says

can be knotted in

if there exists an embedding of

which is not isotopic to

. Traditional knots form the case where

and

.

The Schoenflies theorem states that the circle does not knot in the 2-sphere -- every circle in the 2-sphere is isotopic to the standard circle. Alexander's theorem states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere. In the tame topological category, it's known that the

-sphere does not knot in the

-sphere for all

. This is a theorem of Brown and Mazur. The Alexander horned sphere

Alexander horned sphere

The Alexander horned sphere is a wild embedding of a sphere into space, discovered by . It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:...

is an example of a knotted 2-sphere in the 3-sphere which is not tame. In the smooth category, the

-sphere is known not to knot in the

-sphere provided

. The case

is a long-outstanding problem closely related to the question: does the 4-ball admit an exotic smooth structure

Exotic sphere

In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...

?

Haefliger

André Haefliger

André Haefliger is a Swiss mathematician who works primarily on topology.He studied mathematics in Lausanne. He received his PhD in 1958 from the University of Strasbourg under the supervision of Charles Ehresmann with "Structures feuilletées et cohomologie à valeurs dans un faisceau de...

proved that there are no smooth j-dimensional knots in

provided

, and gave further examples of knotted spheres for all

such that

is called the codimension

Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of

form a group, with group operation given by the connect sum, provided the co-dimension is greater than two.
Haefliger based his work on Smale's h-cobordism theorem. One of Smale's theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory. If one allows topological or PL-isotopies, Zeeman proved that spheres do not knot when the co-dimension is larger than two. See a generalization to manifolds.

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Types of knots

Knots, more generally speaking

See also

Further reading

External links