Bell series: Difference between revisions
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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'', | |||
:<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–Luecke theorem]] states that a knot is determined by its complement. That is, if ''K'' and ''K''′ 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), 371–415. | |||
{{Knot theory|state=collapsed}} | |||
[[Category:Knot theory]] | |||
{{knottheory-stub}} | |||
Revision as of 19:26, 22 January 2014
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 of N,
The knot complement XK is a compact 3-manifold; the boundary of XK 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 complement.
Many knot invariants, 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–Luecke theorem states that a knot is determined by its complement. That is, if K and K′ are two knots with homeomorphic complements then there is a homeomorphism of the three-sphere taking one knot to the other.
See also
Further reading
- C. Gordon and J. Luecke, "Knots are determined by their Complements", J. Amer. Math. Soc., 2 (1989), 371–415.