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}

Read more about this topic:  Figure-eight Knot (mathematics)

Famous quotes containing the word description:

    “God damnit, why must all those journalists be such sticklers for detail? Why, they’d hold you to an accurate description of the first time you ever made love, expecting you to remember the color of the room and the shape of the windows.”
    —Lyndon Baines Johnson (1908–1973)

    “An intentional object is given by a word or a phrase which gives a description under which.”
    —Gertrude Elizabeth Margaret Anscombe (b. 1919)

    “I was here first introduced to Joe.... He was a good-looking Indian, twenty-four years old, apparently of unmixed blood, short and stout, with a broad face and reddish complexion, and eyes, methinks, narrower and more turned up at the outer corners than ours, answering to the description of his race. Besides his underclothing, he wore a red flannel shirt, woolen pants, and a black Kossuth hat, the ordinary dress of the lumberman, and, to a considerable extent, of the Penobscot Indian.”
    —Henry David Thoreau (1817–1862)