Figureeight knot  

 
Common name  Figureeight knot 
Arf invariant  1 
Braid length  4 
Braid no.  3 
Bridge no.  2 
Crosscap no.  2 
Crossing no.  4 
Genus  1 
Hyperbolic volume  2.02988 
Stick no.  7 
Unknotting no.  1 
Conway notation  [22] 
AB notation  4_{1} 
Dowker notation  4, 6, 8, 2 
Last /Next  3_{1} / 5_{1} 
Other  
alternating, hyperbolic, fibered, prime, fully amphichiral, twist 
In knot theory, a figureeight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four. This makes it the knot with the thirdsmallest possible crossing number, after the unknot and the trefoil knot. The figureeight knot is a prime knot.
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Transcription
Contents
Origin of name
The name is given because tying a normal figureeight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot.
Description
A simple parametric representation of the figureeight knot is as the set of all points (x,y,z) where
for t varying over the real numbers (see 2D visual realization at bottom right).
The figureeight knot is prime, alternating, rational with an associated value of 5/2, and is achiral. The figureeight knot is also a fibered knot. This follows from other, less simple (but very interesting) representations of the knot:
(1) It is a homogeneous^{[note 1]} closed braid (namely, the closure of the 3string braid σ_{1}σ_{2}^{−1}σ_{1}σ_{2}^{−1}), and a theorem of John Stallings shows that any closed homogeneous braid is fibered.
(2) It is the link at (0,0,0,0) of an isolated critical point of a realpolynomial map F: R^{4}→R^{2}, so (according to a theorem of John Milnor) the Milnor map of F is actually a fibration. Bernard Perron found the first such F for this knot, namely,
where
Mathematical properties
The figureeight knot has played an important role historically (and continues to do so) in the theory of 3manifolds. Sometime in the midtolate 1970s, William Thurston showed that the figureeight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figureeight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten Dehn surgeries on the figureeight knot resulted in nonHaken, nonSeifertfibered irreducible 3manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston's construction to other knots and links.
The figureeight knot is also the hyperbolic knot whose complement has the smallest possible volume, 2.02988... according to the work of Chun Cao and Robert Meyerhoff. From this perspective, the figureeight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a doublecover of the Gieseking manifold, which has the smallest volume among noncompact hyperbolic 3manifolds.
The figureeight knot and the (−2,3,7) pretzel knot are the only two hyperbolic knots known to have more than 6 exceptional surgeries, Dehn surgeries resulting in a nonhyperbolic 3manifold; they have 10 and 7, respectively. A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figureeight knot is the only one that achieves the bound of 10. A wellknown conjecture is that the bound (except for the two knots mentioned) is 6.
Invariants
The Alexander polynomial of the figureeight knot is
the Conway polynomial is
 ^{[1]}
and the Jones polynomial is
The symmetry between and in the Jones polynomial reflects the fact that the figureeight knot is achiral.
Notes
 ^ A braid is called homogeneous if every generator either occurs always with positive or always with negative sign.
References
 ^ "4_1", The Knot Atlas.
Further reading
 Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4 (2000), 431–449. MR1799796
 Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3manifolds of minimum volume, Inventiones Mathematicae, 146 (2001), no. 3, 451–478. MR1869847
 Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140 (2000), no. 2, 243–282. MR1756996
 Marc Lackenby and Robert Meyerhoff, The maximal number of exceptional Dehn surgeries, arXiv:0808.1176
 Robion Kirby, Problems in lowdimensional topology, (see problem 1.77, due to Cameron Gordon, for exceptional slopes)
 William Thurston, The Geometry and Topology of ThreeManifolds, Princeton University lecture notes (1978–1981).
External links
 "4_1", The Knot Atlas. Accessed: 7 May 2013.
 Weisstein, Eric W. "Figure Eight Knot". MathWorld.