Editing Hyperbolic link

{{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.

==Examples==
[[File:BorromeanRings.svg|thumb|[[Borromean rings]] are a hyperbolic link.]]
*[[Borromean rings]] are hyperbolic.
*Every [[split link|non-split]], [[prime knot|prime]], [[alternating knot|alternating]] link that is not a [[torus link]] is hyperbolic by a result of [[William Menasco]].
*[[Figure-eight knot (mathematics)|4<sub>1</sub> knot]] (the figure-eight knot)
*[[Three-twist knot|5<sub>2</sub> knot]] (the three-twist knot)
*[[Stevedore knot (mathematics)|6<sub>1</sub> knot]] (the stevedore knot)
*[[62 knot|6<sub>2</sub> knot]]
*[[63 knot|6<sub>3</sub> knot]]
*[[74 knot|7<sub>4</sub> knot]]
*[[Perko pair|10 161 knot]] (the "Perko pair" knot)
*[[(−2,3,7) pretzel knot|12n242 knot]]

==See also==
* [[SnapPea]]
* [[Hyperbolic volume (knot)]]

==Further reading==
* [[Colin Adams (mathematician)|Colin Adams]] (1994, 2004) ''The Knot Book'', American Mathematical Society, {{ISBN|0-8050-7380-9}}.
* [[William Menasco]] (1984) "Closed incompressible surfaces in alternating knot and link complements", [[Topology (journal)|Topology]] 23(1):37–44.
* [[William Thurston]] (1978-1981) [[The geometry and topology of three-manifolds]], Princeton lecture notes.

==External links==
*Colin Adams, [https://arxiv.org/abs/math/0309466 ''Handbook of Knot Theory'']

{{Knot theory|state=collapsed}}

[[Category:Knot theory]]
[[Category:Hyperbolic knots and links| ]]
[[Category:3-manifolds]]