---
tag: DYFQ2EQ7J6
kind: definition
name: Hyperbolic knot
msc: [57K10, 57K32]
defines: K47G2Q24DT
uses: [QYEAZTYGPT, GHVW1Z35BS]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_link
---

## Definition

Let $K \subset S^3$ be a tame [[knot|QYEAZTYGPT]].
The knot $K$ is *hyperbolic* if its complement $S^3 \setminus K$ admits a complete Riemannian metric of constant sectional curvature $-1$ with finite volume.

## Remarks

Equivalently, $S^3 \setminus K$ is a [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume, the quotient $\mathbb{H}^3 / \Gamma$ by a discrete torsion-free group $\Gamma$ of isometries.
By Mostow–Prasad rigidity the metric is unique up to isometry, so the hyperbolic volume is a [[knot invariant|GZQ5ZE3X6G]].
Example: the [[figure-eight knot|C01XBGAHC6]] complement is obtained by gluing two regular ideal hyperbolic tetrahedra, so the knot is hyperbolic with volume $2.0298832\ldots$.
Non-example: the complement of a [[torus knot|NS60VV786W]] is Seifert fibered and has no hyperbolic metric.
The 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.
By Thurston, a nontrivial knot is hyperbolic if and only if it is neither a torus knot nor a [[satellite knot|A1E8TDEVJ9]].