# nLab hyperbolic link

Contents

### Context

#### Knot theory

knot theory

Examples/classes:

knot invariants

Related concepts:

category: knot theory

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A link $L \subset S^3$ in the 3-sphere is called a hyperbolic link if its knot complement $S^3 \setminus L$ admits the structure of a hyperbolic metric, hence of a hyperbolic 3-manifold, hence if it is isometric to a quotient space of hyperbolic 3-space by a torsion-free discrete group $\Gamma$ acting by isometries:

$S^3 \setminus L \;\simeq\; \mathbb{H}^3/\Gamma \,.$

(e.g. FKP 17, Def. 2.8)

If the hyperbolic link is in fact a knot (has a single connected component) it is called a hyperbolic knot.

## References

• David Futer, Efstratia Kalfagianni, Jessica S. Purcell, A survey of hyperbolic knot theory (arXiv:1708.07201)