Figure-eight Knot (mathematics) - Description

Description

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

\begin{align} x & = \left(2 + \cos{(2t)} \right) \cos{(3t)} \\ y & = \left(2 + \cos{(2t)} \right) \sin{(3t)} \\ z & = \sin{(4t)} \end{align}

for t varying over the real numbers (see 2D visual realization at bottom right).

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 (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,

where

\begin{align} G(x,y,z,t)=\ & (z(x^2+y^2+z^2+t^2)+x (6x^2-2y^2-2z^2-2t^2), \\ & \ t x \sqrt{2}+y (6x^2-2y^2-2z^2-2t^2)). \end{align}

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