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 (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