Hyperbolic volume (knot)
Encyclopedia
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 knots 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 Weeks
's computer program SnapPea
is the ubiquitous tool used to compute hyperbolic volume of a link.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, the hyperbolic volume of a hyperbolic link
Hyperbolic link
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry...
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 knots with the same volume. A mutation
Mutation (knot theory)
In the mathematical field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose K is a knot given in the form of a knot diagram. Consider a disc D in the projection plane of the diagram whose boundary circle intersects K exactly four times...
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
Knot tabulation
Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. As of May 2008 all prime knots up to 16 crossings have been tabulated.-Beginnings:...
. Jeffrey Weeks
Jeffrey Weeks (mathematician)
Jeffrey Renwick Weeks is an American mathematician, a geometric topologist and cosmologist.-Biography:Weeks received his B.A. from Dartmouth College in 1978, and his Ph.D. in mathematics from Princeton University in 1985, under the supervision of William Thurston...
's computer program SnapPea
SnapPea
SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version as part of his doctoral thesis, supervised by William Thurston. The latest version is 3.0d3...
is the ubiquitous tool used to compute hyperbolic volume of a link.