Editing Knot complement (section)

{{Short description|Complement of a knot in three-sphere}}
{{multiple image|caption_align=center|header_align=center
 | total_width = 250
 | image1 = Blue Unknot.png
 | alt1 = Blue unknot
 | image2 = Torus illustration.png
 | alt2 = Green solid torus
 | footer = The knot complement of the [[unknot]] is [[homeomorphism|homeomorphic]] to a solid torus - notice that while the unknot itself can be represented as a torus, the hole in the unknot corresponds to the solid region of the complement, while the knot itself is the hole in the complement. This is connected to the trivial [[Heegaard decomposition]] of the 3-sphere into two solid tori.
}}

In [[mathematics]], the '''knot complement''' of a [[tame knot]] ''K'' is the space where the knot is not. If a knot is embedded in the [[3-sphere]], then the complement is the 3-sphere minus the space near 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 [[tubular 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 the [[3-sphere]].   Context is needed to determine the usage.  There are analogous definitions for the '''[[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.

Knot complements are [[Haken manifolds]].<ref>{{cite book |last=Jaco |first=William |author-link=William Jaco |date=1980 |title=Lectures on Three-Manifold Topology |url=https://bookstore.ams.org/cbms-43 |publisher=AMS |page=42 |isbn=978-1-4704-2403-9}}</ref> More generally complements of [[Link_(knot_theory)|links]] are Haken manifolds.