Hyperbolic knot

Remarks

Equivalently, $S^3 \setminus K$ is a hyperbolic 3-manifold of finite volume, the quotient $\mathbb{H}^3 / \Gamma$ by a discrete torsion-free group $\Gamma$ of isometries. By Mostow–Prasad rigidity the metric is unique up to isometry, so the hyperbolic volume is a knot invariant. Example: the figure-eight knot complement is obtained by gluing two regular ideal hyperbolic tetrahedra, so the knot is hyperbolic with volume $2.0298832\ldots$. Non-example: the complement of a torus knot is Seifert fibered and has no hyperbolic metric. The complement of the unknot is an open solid torus, whose fundamental group $\Z$ cannot be a lattice in the isometry group of $\mathbb{H}^3$; so the unknot is not hyperbolic. By Thurston, a nontrivial knot is hyperbolic if and only if it is neither a torus knot nor a satellite knot.