Figure-eight knot (mathematics)

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In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four, the smallest possible except for the unknot and trefoil knot.

Contents 1 Origin of name 2 Description 3 Importance in mathematics 4 References

[edit] Origin of name

The name is given because joining the ends of a string with a normal figure-of-eight knot tied in it, in the most natural way, gives a model of the mathematical knot.

[edit] Description

A simple representation of the figure-eight knot is as the set of all points (x,y,z) where

x = (2 + cos(2t)) cos(3t)

y = (2 + cos(2t)) sin(3t)

z = sin(4t)

for some real value of t. The knot is alternating, rational with an associated value of 5/2, and is achiral.

The figure-eight knot is also a fibered knot. This follows from other, less simple (but very interesting) representations of the knot. (1) It is a homogeneous closed braid (namely, the closure of the 3-string braid σ₁σ₂^-1σ₁σ₂^-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 real-polynomial map F:R⁴→R², 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,

F(x,y,z,t)=G(x,y,z²-t²,2zt),

where

G(x,y,z,t)=(z(x²+y²+z²+t²)+x(6x²-2y²-2z²-2t²), sqrt(2)tx+y(6x²-2y²-2z²-2t²)).

[edit] Importance in mathematics

The figure-eight knot has played an important role historically (and continues to do so) in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jorgensen, working independently of each other, had earlier shown that the figure-eight 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 figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston's construction to other knots and links.

The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume, 2.02988... by work of Chun Cao and Robert Meyerhoff. From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot.

[edit] References

• Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae, 146 (2001), no. 3, 451--478.
• William Thurston, The Geometry and Topology of Three-Manifolds, Princeton University lecture notes (1978-1981).