View source for 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|mathematics]], a '''hyperbolic link''' is a [[Link (knot theory)|link]] in the [[3-sphere|3-sphere]] with [[Knot complement|complement]] that has a complete Riemannian metric of constant negative [[Curvature|curvature]], i.e. has a [[Hyperbolic geometry|hyperbolic geometry]]. A '''hyperbolic knot''' is a hyperbolic link with one [[Connected space|component]].

As a consequence of the work of [[Biography:William Thurston|William Thurston]], it is known that every knot is precisely one of the following: hyperbolic, a [[Torus knot|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|hyperbolic Dehn surgery]] theorem, performing [[Dehn surgery|Dehn surgeries]] on a hyperbolic link enables one to obtain many more [[Hyperbolic 3-manifold|hyperbolic 3-manifold]]s.

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

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

==Further reading==
* [[Biography:Colin Adams (mathematician)|Colin Adams]] (1994, 2004) ''The Knot Book'', American Mathematical Society, {{ISBN|0-8050-7380-9}}.
* [[Biography:William Menasco|William Menasco]] (1984) "Closed incompressible surfaces in alternating knot and link complements", Topology 23(1):37–44.
* [[Biography:William Thurston|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:Hyperbolic knots and links| ]]
[[Category:3-manifolds]]

{{Sourceattribution|Hyperbolic link}}