Hyperbolic volume

The volume is necessarily a finite real number, and is a topological invariant of the link.

[1] As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.

[2] A hyperbolic link is a link in the 3-sphere whose complement (the space formed by removing the link from the 3-sphere) can be given a complete Riemannian metric of constant negative curvature, giving it the structure of a hyperbolic 3-manifold, a quotient of hyperbolic space by a group acting freely and discontinuously on it.

The components of the link will become cusps of the 3-manifold, and the manifold itself will have finite volume.

[5] Thurston and Jørgensen proved that the set of real numbers that are hyperbolic volumes of 3-manifolds is well-ordered, with order type ωω.