 Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3dimensional Euclidean space, R^{3}, considered up to continuous deformations (isotopies). 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. 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 S^{j} in S^{n}, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory.
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Types of knots
The simplest knot, called the unknot, is a round circle embedded in R^{3}. In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot (3_{1} in the table), the figureeight knot (4_{1}) and the cinquefoil knot (5_{1}).
Several knots, possibly tangled together, are called links. Knots are links with a single component.
Often mathematicians prefer to consider knots embedded into the 3sphere, S^{3}, rather than R^{3} since the 3sphere is compact. The 3sphere is equivalent to R^{3} with a single point added at infinity (see onepoint 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 3sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. Knots that are not tame are called wild and can have pathological behavior. In knot theory and 3manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.
Given a knot in the 3sphere, the knot complement is all the points of the 3sphere 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 3manifold theory.
The JSJ decomposition and Thurston's hyperbolization theorem reduces the study of knots in the 3sphere to the study of various geometric manifolds via splicing or satellite operations. In the pictured knot, the JSJdecomposition splits the complement into the union of three manifolds: two trefoil complements and the complement of the Borromean rings. The trefoil complement has the geometry of , while the Borromean rings complement has the geometry of H^{3}.
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 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 = S^{1} and or M = S^{3}.
The Schoenflies theorem states that the circle does not knot in the 2sphere  every circle in the 2sphere is isotopic to the standard circle. Alexander's theorem states that the 2sphere does not smoothly (or PL or tame topologically) knot in the 3sphere. In the tame topological category, it's known that the nsphere does not knot in the n + 1sphere for all n. This is a theorem of Brown and Mazur. The Alexander horned sphere is an example of a knotted 2sphere in the 3sphere which is not tame. In the smooth category, the nsphere is known not to knot in the n + 1sphere provided . The case n = 3 is a longoutstanding problem closely related to the question: does the 4ball admit an exotic smooth structure?
Haefliger proved that there are no smooth jdimensional knots in S^{n} 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. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of S^{j} in S^{n} form a group, with group operation given by the connect sum, provided the codimension is greater than two. Haefliger based his work on Smale's hcobordism theorem. One of Smale's theorems is that when one deals with knots in codimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than codimension 2 knot theory. If one allows topological or PLisotopies, Zeeman proved that spheres do not knot when the codimension is larger than two. See a generalization to manifolds.
See also
Further reading
 David W. Farmer & Theodore B. Stanford, Knots and Surfaces: A Guide to Discovering Mathematics, 1995.
 Colin C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W. H. Freeman & Company, March 1994.
 Charles Livingstone, Knot Theory, The Mathematical Association of America, September 1996.
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