nLab hyperbolic link




A link LS 3L \subset S^3 in the 3-sphere is called a hyperbolic link if its knot complement S 3LS^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 Γ\Gamma acting by isometries:

S 3L 3/Γ. 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.


  • 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

Last revised on December 12, 2019 at 07:41:22. See the history of this page for a list of all contributions to it.