Description
A simple parametric representation of the figure-eight knot is as the set of all points (x,y,z) where
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
Read more about this topic: Figure-eight Knot (mathematics)
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