In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed -- there are no ends to tie or untie on a mathematical knot. KNOT (1450 AM) is a commercial Classic Country music Radio station in Prescott Arizona, broadcasting to the Flagstaff - Prescott 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 higher dimensional manifolds. The branch of mathematics that studies knots is known as knot theory. In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces
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Types of knots
The simplest knot, called the unknot, is a round circle embedded in R3. The unknot arises in the mathematical theory of knots. Intuitively the unknot is a closed loop of rope without a Knot in it In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot (31 in the table), the figure-eight knot (41) and the cinquefoil knot (51). In Knot theory, the trefoil knot is the simplest nontrivial knot. In Knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four The cinquefoil knot, also known as Solomon's seal knot, and 51 in most tables is a (52- Torus knot with five crossings
Several knots, possibly tangled together, are called links. In Mathematics, a link is a collection of knots which do not intersect but which may be linked (or knotted together Knots are links with a single component.
Often mathematicians prefer to consider knots embedded into the 3-sphere, S3, rather than R3 since the 3-sphere is compact. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. 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, 
, into the 3-sphere. In Mathematics, a solid torus is a Topological space Homeomorphic to S^1 \times D^2 i A knot is tame if and only it can be represented as a finite closed polygonal chain. A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of Line segments More formally a polygonal Knots that are not tame are called wild and can have pathological behavior. In Mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive In knot theory and 3-manifold theory, often the adjective "tame" is omitted. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions Smooth knots, for example, are always tame.
Given a knot in the 3-sphere, the knot complement is all the points of the 3-sphere not contained in the knot. 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 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. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions
The JSJ decomposition and Thurston's hyperbolization theorem reduces the study of knots in the 3-sphere to the study of various geometric manifolds via splicing or satellite operations. In the pictured knot, the JSJ-decomposition splits the complement into the union of three manifolds: two trefoil complements and the complement of the Borromean rings. In Mathematics, the JSJ decomposition, also known as the toral decomposition, is a Topological construct given by the following theorem Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures In the mathematical theory of knots, a satellite knot is a knot which contains an incompressible, non- Boundary parallel Torus in its In Knot theory, the trefoil knot is the simplest nontrivial knot. In Mathematics, the Borromean rings consist of three topological Circles which are linked and form a Brunnian link, i The trefoil complement has the geometry of 
, while the Borromean rings complement has the geometry of H3.
Knots, more generally speaking
In contemporary mathematics the term knot is sometimes used to describe a more general phenomena related to embeddings. Given a manifold M with a submanifold N, one sometimes says N can be knotted in M if there exists an embedding of N in M which is not isotopic to N. Traditional knots form the case where N = S1 and M = S3.
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. In Mathematics, the Jordan–Schönflies theorem, or simply the Schönflies theorem, of Geometric topology is a sharpening of the Jordan curve theorem Alexander's theorem states that the 2-sphere does not knot in the 3-sphere. In the tame topological category, it's known that the n-sphere does not knot in the n + 1-sphere for all n. This is a theorem of Brown and Mazur. The Alexander horned sphere is an example of a knotted 2-sphere in the 3-sphere which is not tame. The Alexander horned sphere is one of the most famous pathological examples in Mathematics discovered in 1924 by J In the smooth category, the n-sphere is known not to knot in the n + 1-sphere provided 
. The case n = 3 is a long-outstanding problem closely related to the question: does the 4-ball admit an exotic smooth structure?
Haefliger proved that there are no smooth j-dimensional knots in Sn provided 2n − 3j − 3 > 0, and gave further examples of knotted spheres for all 
such that 2n − 3j − 3 = 0. n − j is called the codimension of the knot. In Mathematics, codimension is a basic geometric idea that applies to Subspaces in Vector spaces and more generally to Submanifolds in Manifolds An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of Sj in Sn 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 also
External links
Further Reading
- David W. In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces This article contains a list of mathematical knots and links. In Knot theory, a prime knot is a knot which is in a certain sense indecomposable A rosette is a round stylized flower design used extensively in sculptural objects from antiquity. Farmer & Theodore B. Stanford, Knots and Surfaces: A Guide to Discovering Mathematics, 1995.
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