- Hyperbolic link
In
mathematics , a hyperbolic link is a link in the3-sphere with complement that has a completeRiemannian metric of constant negativecurvature , i.e. has ahyperbolic 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, atorus knot , or asatellite 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 morehyperbolic 3-manifold s.Examples
*Every non-split, prime, alternating link that is not a
torus link is hyperbolic by a result ofWilliam Menasco .ee also
*
SnapPea
*hyperbolic volume (knot) 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).Further reading
*Colin Adams, [http://front.math.ucdavis.edu/math.GT/0309466 "Hyperbolic knots" (arXiv preprint)]
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