nLab hyperbolic link

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.

Related concepts

• hyperbolic 3-manifold

• volume conjecture

References

• David Futer, Efstratia Kalfagianni, Jessica S. Purcell, A survey of hyperbolic knot theory, Springer Proceedings in Mathematics & Statistics, vol. 284 (2019), 1-30 (arXiv:1708.07201)

See also

• Wikipedia, Hyperbolic link