Hyperbolic link

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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.

Contents 1 Examples 2 See also 3 References 4 Further reading

[edit] 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.

[edit] See also

• SnapPea
• hyperbolic volume (knot)

[edit] References

• Colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9
• 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 3-manifolds, Princeton lecture notes (1978-1981).

[edit] Further reading

• Colin Adams, Hyperbolic knots (arXiv preprint)