Hyperbolic volume (knot)

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In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is simply the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily finite. The hyperbolic volume of a non-hyperbolic knot is often defined to be zero. By Mostow rigidity, the volume is a topological invariant of the link.

It is known that there are only finitely many hyperbolic links with the same volume. A mutation of a hyperbolic knot will have the same volume, so it is possible to concoct examples with the same volume. In practice, hyperbolic volume has proven very effective in distinguishing knots, utilized in some of the extensive efforts at knot tabulation. Jeffrey Week's computer program SnapPea is the ubiquitous tool used to compute hyperbolic volume of a link.