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| {{Merge from|knots and graphs|discuss=Talk:Knot (mathematics)#Merger proposal|date=October 2011}}
| | I'm Nߋna and I live in Wermɑtѕwil. <br>I'm intereѕted in Playwriting, Meteorology and Nοrwegian art. I like to travel and reading fantɑsy.<br><br>My homepage ... [http://champv.dx.am/blomsomtahiti786582 blomsom so] |
| [[Image:Knot table.svg|thumb|350px|right|A table of all [[prime knot]]s with seven [[Crossing number (knot theory)|crossings]] or fewer (not including mirror images).]]
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| [[Image:Example of Knots.svg|180px|thumb|Overhand knot becomes a trefoil knot by joining the ends.]]
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| In [[mathematics]], a '''knot''' is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], '''R'''<sup>3</sup>, considered up to continuous deformations ([[homotopy|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 <math>S^j</math> in <math>S^n</math>, especially in the case <math>j=n-2</math>. The branch of mathematics that studies knots is known as [[knot theory]].
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| ==Formal definition==
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| A knot is an [[embedding#General topology|embedding]] of the [[circle]] ([[n-sphere|S<sub>1</sub>]]) into [[three-dimensional space|three-dimensional]] [[Euclidean space]] ('''E'''<sup>3</sup>).<ref>{{cite book |last=Armstrong |first=M. A.| year=1983 |origyear=1979 |location=New York |publisher=Springer-Verlag |series=Undergraduate Texts in Mathematics |title=Basic Topology |isbn=0-387-90839-0 |page=213 |ref=harv}}</ref> Two knots are defined to be equivalent if there is an [[ambient isotopy]] between them{{citation needed|reason=Armstrong uses a looser equivalence on p. 214, only requiring an ambient homeomorphism. He then goes on to discuss why it would be a good idea to restrict this to ambient isotopy. I know this latter definition is standard, but I Armstrong doesn't directly support that and I don't have a citation handy|date=December 2011}}.
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| ===Tame vs. wild knots===
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| A ''polygonal'' knot is a knot whose [[image (mathematics)|image]] in '''E'''<sup>3</sup> is the [[union (set theory)|union]] of a [[finite set]] of [[line segments]].<ref name="Armstrong">Armstrong (1983), p.215.</ref> A ''tame'' knot is any knot equivalent to a polygonal knot.<ref name="Armstrong"/> Knots which are not tame are called ''[[Wild knot|wild]]''.<ref>{{mathworld |title=Wild Knot |id=WildKnot}}</ref>
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| ==Types of knots==
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| {{unreferenced section|date=December 2011}}
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| The simplest knot, called the [[unknot]] or trivial knot, is a round circle embedded in [[Euclidean space|'''R'''<sup>3</sup>]]. In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the [[trefoil knot]] (3<sub>1</sub> in the table), the [[figure-eight knot (mathematics)|figure-eight knot]] (4<sub>1</sub>) and the [[cinquefoil knot]] (5<sub>1</sub>).
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| [[Image:Untying a knot.svg|thumb|300px|A knot can be untied if the loop is broken.]]
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| Several knots, linked or tangled together, are called [[link (knot theory)|links]]. Knots are links with a single component.
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| Often mathematicians prefer to consider knots embedded into the [[3-sphere]], '''S'''<sup>3</sup>, rather than '''R'''<sup>3</sup> since the 3-sphere is [[compact space|compact]]. The 3-sphere is equivalent to '''R'''<sup>3</sup> with a single point added at infinity (see [[one-point compactification]]).
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| [[Image:Wild knot.svg|thumb|200px|A wild knot.]]
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| A knot is [[tame knot|tame]] if it can be "thickened up", that is, if there exists an extension to an embedding of the [[solid torus]], <math>S^1 \times D^2</math>, into the 3-sphere. A knot is tame if and only if it can be represented as a finite [[polygonal chain|closed polygonal chain]]. Knots that are not tame are called wild and can have [[pathological (mathematics)|pathological]] behavior. In knot theory and [[3-manifold]] theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.
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| Given a knot in the 3-sphere, the [[knot complement]] 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-manifolds|3-manifold theory]].
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| [[Image:Knot with borromean rings in jsj decomp small.png|thumb|150px|right|A knot whose complement has a non-trivial JSJ decomposition.]]
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| The [[JSJ decomposition]] and [[geometrization conjecture|Thurston's hyperbolization theorem]] reduces the study of knots in the 3-sphere to the study of various geometric manifolds via ''splicing'' or ''[[satellite knot|satellite operations]]''. In the pictured knot, the JSJ-decomposition splits the complement into the union of three manifolds: two [[trefoil knot|trefoil complements]] and the complement of the [[Borromean rings]]. The trefoil complement has the geometry of <math>H^2 \times R</math>, while the Borromean rings complement has the geometry of <math>H^3</math>.
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| ==Generalization==
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| {{refimprove section|date=December 2011}}
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| In contemporary mathematics the term ''knot'' is sometimes used to describe a more general phenomenon related to embeddings{{citation needed|date=December 2011}}. Given a manifold <math>M</math> with a submanifold <math>N</math>, one sometimes says <math>N</math> can be knotted in <math>M</math> if there exists an embedding of <math>N</math> in <math>M</math> which is not isotopic to <math>N</math>. Traditional knots form the case where <math>N=S^1</math> and <math>M=\mathbb R^3</math> or <math>M=S^3</math>.
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| 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 <math>n</math>-sphere does not knot in the <math>n+1</math>-sphere for all <math>n</math>. 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. In the smooth category, the <math>n</math>-sphere is known not to knot in the <math>n+1</math>-sphere provided <math>n \neq 3</math>. The case <math>n=3</math> is a long-outstanding problem closely related to the question: does the 4-ball admit an [[exotic sphere|exotic smooth structure]]?
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| [[André Haefliger|Haefliger]] proved that there are no smooth j-dimensional knots in <math>S^n</math> provided <math>2n-3j-3>0</math>, and gave further examples of knotted spheres for all <math>n > j \geq 1</math> such that <math>2n-3j-3=0</math>. <math>n-j</math> is called the [[codimension]] of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of <math>S^j</math> in <math>S^n</math> form a group, with group operation given by the connect sum, provided the co-dimension is greater than two.
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| 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 [[Whitney embedding theorem#Isotopy versions|generalization to manifolds]].
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| ==See also==
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| [[File:Celtic 7 4 Knot.jpg|thumb|7<sub>4</sub> [[pretzel link]] knot]]
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| *[[List of mathematical knots and links]]
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| *[[Rosette (design)]]
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| == References ==
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| {{reflist}}
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| == Further reading ==
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| *{{cite book |first=David W. |last=Farmer |first2=Theodore B. |last2=Stanford |title=Knots and Surfaces: A Guide to Discovering Mathematics |year=1995 }}
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| *{{cite book |first=Colin C. |last=Adams |title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=W. H. Freeman & Company |date=March 1994 }}
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| *{{cite book |first=Charles |last=Livingstone |title=Knot Theory |publisher=The Mathematical Association of America |date=September 1996 }}
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| ==External links==
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| {{commons category|Knots (knot theory)}}
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| *{{Knot Atlas|Main_Page}}
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| *{{Wayback|url=http://www.map.him.uni-bonn.de/index.php/High_codimension_embeddings:_classification|title="Classification of embeddings", ''The Manifold Atlas Project''|date=20120304102524}}
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| {{Knot theory|state=collapsed}}
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| [[Category:Knots (knot theory)| ]]
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I'm Nߋna and I live in Wermɑtѕwil.
I'm intereѕted in Playwriting, Meteorology and Nοrwegian art. I like to travel and reading fantɑsy.
My homepage ... blomsom so