In knot theory, a figureeight knot (also called Listing's knot) is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot. The figureeight knot is a prime knot.
Contents
Origin of name
The name is given because tying a normal figureofeight 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.
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 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.
Full article ▸
