# Hyperbolic link

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
- 5₂ knot
- 6₁ knot
- 6₂ knot
- 6₃ knot
- 7₄ knot
- 10 161 knot
- 12n242 knot

## See also

## 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,
*Hyperbolic knots*(arXiv preprint)

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