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.
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
- 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, Handbook of Knot Theory
This article is adapted from “Hyperbolic link” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.