Hyperbolic link

{{Short description|Type of mathematical link}} [[File:Blue Figure-Eight Knot.png|thumb|4₁ knot]]

In mathematics, a '''hyperbolic link''' is a link in the 3-sphere with 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 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.{{Clarify|reason=what heuristic?|date=January 2026}}

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

==Examples== [[File:BorromeanRings.svg|thumb|Borromean rings are a hyperbolic link.]] *Borromean rings are hyperbolic. *Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco. *4₁ knot (the figure-eight knot) *5₂ knot (the three-twist knot) *6₁ knot (the stevedore knot) *6₂ knot *6₃ knot *7₄ knot *10 161 knot (the "Perko pair" knot) *12n242 knot

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

==Further reading== * 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 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