Hyperbolic link
Type of mathematical link
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/220px-Blue_Figure-Eight_Knot.png)
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. A hyperbolic knot is a hyperbolic link with one component.
As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links.
As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds.
Examples
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/BorromeanRings.svg/220px-BorromeanRings.svg.png)
- Borromean rings are hyperbolic.
- Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.
- 41 knot (the figure-eight knot)
- 52 knot (the three-twist knot)
- 61 knot (the stevedore knot)
- 62 knot
- 63 knot
- 74 knot
- 10 161 knot (the "Perko pair" knot)
- 12n242 knot
See also
- SnapPea
- Hyperbolic volume (knot)
Further reading
- Colin Adams (1994, 2004) The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9.
- William Menasco (1984) "Closed incompressible surfaces in alternating knot and link complements", Topology 23(1):37–44.
- William Thurston (1978-1981) The geometry and topology of three-manifolds, Princeton lecture notes.
External links
- Colin Adams, Handbook of Knot Theory
- v
- t
- e
Knot theory (knots and links)
- Figure-eight (41)
- Three-twist (52)
- Stevedore (61)
- 62
- 63
- Endless (74)
- Carrick mat (818)
- Perko pair (10161)
- Conway knot (11n34)
- Kinoshita–Terasaka knot (11n42)
- (−2,3,7) pretzel (12n242)
- Whitehead (52
1) - Borromean rings (63
2) - L10a140
- Composite knots
- Granny
- Square
- Knot sum
and operations
- Alexander–Briggs notation
- Conway notation
- Dowker–Thistlethwaite notation
- Flype
- Mutation
- Reidemeister move
- Skein relation
- Tabulation
Category
Commons