Editing Hyperbolic link (section)

{{Short description|Type of mathematical link}}
[[File:Blue Figure-Eight Knot.png|thumb|[[Figure-eight knot (mathematics)|4<sub>1</sub> knot]]]]

In [[mathematics]], a '''hyperbolic link''' is a [[link (knot theory)|link]] in the [[3-sphere]] with [[knot complement|complement]] that has a complete [[Riemannian metric]] of constant negative [[curvature]], i.e. has a [[hyperbolic geometry]]. A '''hyperbolic knot''' is a hyperbolic link with one [[connected space|component]].

As a consequence of the work of [[William Thurston]], it is known that every knot is precisely one of the following: hyperbolic, a [[torus knot]], or a [[satellite knot]]. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links. 

As a consequence of Thurston's [[hyperbolic Dehn surgery]] theorem, performing [[Dehn surgery|Dehn surgeries]] on a hyperbolic link enables one to obtain many more [[hyperbolic 3-manifold]]s.