Hyperbolic link
Type of mathematical link
title: "Hyperbolic link" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["knot-theory", "hyperbolic-knots-and-links", "3-manifolds"] description: "Type of mathematical link" topic_path: "general/knot-theory" source: "https://en.wikipedia.org/wiki/Hyperbolic_link" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Type of mathematical link ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/05/Blue_Figure-Eight_Knot.png" caption="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.
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
::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/c2/BorromeanRings.svg" caption="[[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
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::