Hyperbolic knot

Summary

A hyperbolic knot is a knot whose complement in the $3$-sphere carries a complete hyperbolic metric of finite volume. By Mostow–Prasad rigidity this metric is unique, so its volume and other geometric quantities are knot invariants. Thurston showed that every nontrivial knot is a torus knot, a satellite knot or a hyperbolic knot, and exactly one of these. Most knots in the tables are hyperbolic.