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  "body": "\n## Definition\n\nLet $K \\subset S^3$ be a tame [[knot|QYEAZTYGPT]].\nThe knot $K$ is *hyperbolic* if its complement $S^3 \\setminus K$ admits a complete Riemannian metric of constant sectional curvature $-1$ with finite volume.\n\n## Remarks\n\nEquivalently, $S^3 \\setminus K$ is a [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume, the quotient $\\mathbb{H}^3 / \\Gamma$ by a discrete torsion-free group $\\Gamma$ of isometries.\nBy Mostow–Prasad rigidity the metric is unique up to isometry, so the hyperbolic volume is a [[knot invariant|GZQ5ZE3X6G]].\nExample: the [[figure-eight knot|C01XBGAHC6]] complement is obtained by gluing two regular ideal hyperbolic tetrahedra, so the knot is hyperbolic with volume $2.0298832\\ldots$.\nNon-example: the complement of a [[torus knot|NS60VV786W]] is Seifert fibered and has no hyperbolic metric.\nThe complement of the [[unknot|1TQ06R4Q66]] 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.\nBy Thurston, a nontrivial knot is hyperbolic if and only if it is neither a torus knot nor a [[satellite knot|A1E8TDEVJ9]].\n",
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