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Hyperbolic Knot

A hyperbolic knot is a Knot that has a complement that can be given a metric of constant curvature $-1$. The only Knots which are not hyperbolic are Torus Knots and Satellite Knots (including Composite Knots), as proved by Thurston in 1978. Therefore, all but six of the Prime Knots with 10 or fewer crossings are hyperbolic. The exceptions with nine or fewer crossings are 03-001 (the $(3,2)$-Torus Knot), 05-001, 07-001, 08-019 (the $(4,3)$-Torus Knot), and 09-001.

Almost all hyperbolic knots can be distinguished by their hyperbolic volumes (exceptions being 05-002 and a certain 12-crossing knot; see Adams 1994, p. 124). It has been conjectured that the smallest hyperbolic volume is 2.0298..., that of the Figure-of-Eight Knot.

Mutant Knots have the same hyperbolic knot volume.


Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 119-127, 1994.

Adams, C.; Hildebrand, M.; and Weeks, J. ``Hyperbolic Invariants of Knots and Links.'' Trans. Amer. Math. Soc. 326, 1-56, 1991.

mathematica.gif Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.

© 1996-9 Eric W. Weisstein