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In [[mathematics]], the '''knot complement''' of a [[tame knot]] ''K'' is the three-dimensional space surrounding the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the [[3-sphere]]). Let ''N'' be a thickened neighborhood of ''K''; so ''N'' is a [[solid torus]]. The knot complement is then the [[complement (set theory)|complement]] of ''N'',
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:<math>X_K = M - \mbox{interior}(N).</math>
 
The knot complement ''X<sub>K</sub>'' is a [[compact space|compact]] [[3-manifold]]; the boundary of ''X<sub>K</sub>'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-[[torus]].  Sometimes the ambient manifold ''M'' is understood to be [[3-sphere]].  Context is needed to determine the usage. There are analogous definitions of '''[[link (knot theory)|link]] complement'''.
 
Many [[knot invariant]]s, such as the [[knot group]], are really invariants of the complement of the knot.  When the ambient space is the three-sphere no information is lost: the [[Gordon&ndash;Luecke theorem]] states that a knot is determined by its complement. That is, if ''K'' and ''K''&prime; are two knots with [[homeomorphic]] complements then there is a homeomorphism of the three-sphere taking one knot to the other.
 
==See also==
[[Seifert surface]]
 
==Further reading==
* C. Gordon and J. Luecke, "Knots are determined by their Complements", ''[[Journal of the American Mathematical Society|J. Amer. Math. Soc.]]'', '''2''' (1989), 371&ndash;415.
 
{{Knot theory|state=collapsed}}
 
[[Category:Knot theory]]
 
{{knottheory-stub}}

Latest revision as of 17:56, 22 September 2014

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