A hyperbolic knot is a knot that has a complement that can be given a metric of constant curvature  . All hyperbolic
knots are prime knots (Hoste et
al. 1998).
A prime knot on 10 or fewer crossings can be tested in Mathematica to see if it is hyperbolic using KnotData[knot,
"Hyperbolic"].
Knots which are not hyperbolic are either torus knots or satellite knots, as proved by Thurston in 1978. Of the prime
knots with 16 or fewer crossings, all but 32 are hyperbolic. Of these 32, 12 are
torus knots and the remaining 20 are satellites
of the trefoil knot (Hoste et
al. 1998). The nonhyperbolic knots with nine or fewer crossings are all torus
knots, including  (the  -torus knot),  ,  ,  (the  -torus
knot), and  , the first few of which are illustrated
above.
The following table gives the number of nonhyperbolic and hyperbolic knots of  crossing starting with  .
| type | Sloane | counts | | torus | A051764 | 1, 0, 1, 0, 1, 1, 1, 1, 1, 0,
1, 1, 2, 1, ... | | satellite | A051765 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ... | | nonhyperbolic | A052407 | 1, 0, 1, 0, 1, 1, 1, 1, 1, 0,
3, 3, 8, 11, ... | | hyperbolic | A052408 | 0, 1, 1, 3, 6, 20, 48, 164, 551, 2176, 9985, 46969, 253285, 1388694,
... |
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 eight knot.
Mutant knots have the same hyperbolic
knot volume.
The knot symmetry group of a hyperbolic knot must be either a finite cyclic
group or a finite dihedral group
(Riley 1979, Kodama and Sakuma 1992, Hoste et al. 1998).
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.
Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots."
Math. Intell. 20, 33-48, Fall 1998.
Kodama K. and Sakuma, M. "Symmetry Groups of Prime Knots Up to 10 Crossings." In Knot 90, Proceedings of the International Conference on Knot Theory
and Related Topics, Osaka, Japan, 1990 (Ed. A. Kawauchi.) Berlin: de
Gruyter, pp. 323-340, 1992.
Riley, R. "An Elliptic Path from Parabolic Representations to Hyperbolic Structures." In Topology of Low-Dimensional Manifolds, Proceedings, Sussex 1977
(Ed. R. Fenn). New York: Springer-Verlag, pp. 99-133, 1979.
Sloane, N. J. A. Sequences A051764, A051765, A052407, A052408 in "The On-Line Encyclopedia of Integer Sequences."
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