---
tag: K47G2Q24DT
kind: concept
name: Hyperbolic knot
msc: [57K10, 57K32]
specializes: [QYEAZTYGPT]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_link
---

## Summary

A hyperbolic knot is a [[knot|QYEAZTYGPT]] 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|NS60VV786W]], a [[satellite knot|A1E8TDEVJ9]] or a hyperbolic knot, and exactly one of these.
Most knots in the tables are hyperbolic.

## Explanations

### L1 human

Remove a knot from space and study the space that is left over.
For a hyperbolic knot, that leftover space can be given the geometry of the hyperbolic plane, raised to three dimensions, in which triangles have angle sums below $180$ degrees.
This geometry is rigid, so its total volume is a number attached to the knot.
The [[figure-eight knot|C01XBGAHC6]] is hyperbolic, and its complement has volume about $2.0299$.
The trefoil is not hyperbolic, because it is a torus knot.