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.
- 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
- Colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9.[year missing]
- William Menasco, "Closed incompressible surfaces in alternating knot and link complements", Topology 23 (1984), no. 1, 37--44.
- William Thurston, The geometry and topology of three-manifolds, Princeton lecture notes (1978-1981).
- Colin Adams, Hyperbolic knots (arXiv preprint)
|This Knot theory-related article is a stub. You can help Wikipedia by expanding it.|