In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.
Origin of name
The name is given because tying a normal figure-eight 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 figure-eight knot is as the set of all points where for t varying over the real numbers. The figure-eight knot is prime, 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 representations of the knot: It is a homogeneousclosed braid, and a theorem of John Stallings shows that any closed homogeneous braid is fibered. It is the link at of an isolated critical point of a real-polynomial map F: R4→R2, so the Milnor map of F is actually a fibration. Bernard Perron found the first such F for this knot, namely, where
Mathematical properties
The figure-eight knot has played an important role historically 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. This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but tenDehn 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... according to the work of Chun Cao and Robert Meyerhoff. From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, which has the smallest volume among non-compact hyperbolic 3-manifolds. The figure-eight knot and the pretzel knot are the only two hyperbolic knots known to have more than 6 exceptional surgeries, Dehn surgeries resulting in a non-hyperbolic 3-manifold; 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 figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound is 6.
