Hyperbolic volume
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number. 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;[1] as a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.[2]
There are only finitely many hyperbolic knots with the same volume.[2] A mutation of a hyperbolic knot will have the same volume,[3] so it is possible to concoct examples with the same volume; indeed, there are arbitrarily large finite sets of distinct knots with equal volumes.[2] 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.[1]
More generally, the hyperbolic volume may be defined for any hyperbolic 3-manifold. The Weeks manifold has the smallest possible volume of any closed manifold (a manifold that, unlike link complements, has no cusps); its volume is approximately 0.9427.[4]
List
- Figure-eight knot = 2.0298832 (sequence A091518 in OEIS)
- Three-twist knot = 2.82812
- Stevedore knot (mathematics) = 3.16396
- 6₂ knot = 4.40083
- Endless knot = 5.13794
- Perko pair = 5.63877
- 6₃ knot = 5.69302
References
- ↑ 1.0 1.1 Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey (1991), "Hyperbolic invariants of knots and links", Transactions of the American Mathematical Society 326 (1): 1–56, doi:10.2307/2001854, MR 994161.
- ↑ 2.0 2.1 2.2 Wielenberg, Norbert J. (1981), "Hyperbolic 3-manifolds which share a fundamental polyhedron", Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton, N.J.: Princeton Univ. Press, pp. 505–513, MR 624835.
- ↑ Ruberman, Daniel (1987), "Mutation and volumes of knots in S3", Inventiones Mathematicae 90 (1): 189–215, doi:10.1007/BF01389038, MR 906585.
- ↑ Gabai, David; Meyerhoff, Robert; Milley, Peter (2009), "Minimum volume cusped hyperbolic three-manifolds", Journal of the American Mathematical Society 22 (4): 1157–1215, arXiv:0705.4325, doi:10.1090/S0894-0347-09-00639-0, MR 2525782.
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