Hyperbolic link

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

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

See also

• SnapPea
• Hyperbolic volume (knot)

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, Handbook of Knot Theory