Everything about Figure-eight Knot Mathematics totally explained
In
knot theory, a
figure-eight 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.
Origin of name
The name is given because tying a normal
figure-of-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 representation of the figure-eight knot is as the set of all points (
x,
y,
z) where
» x = (2 + cos(2
t)) cos(3
t)
» y = (2 + cos(2
t)) sin(3
t)
» z = sin(4
t)
for some real value of
t. The knot is
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 σ
1σ
2-1σ
1σ
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 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
»
Mathematical properties
The figure-eight knot has played an important role historically (and continues to do so) 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. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight 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 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... by 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 and the
(-2, 3, 7) 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've 10 and 7, respectively. There exists a universal upper bound for the number of exceptional surgeries on any hyperbolic knot. The current best bound is 12, due independently to Ian Agol and Marc Lackenby, utilizing the
geometrization conjecture. A well-known conjecture is that the bound (except for the two knots mentioned) is 6.
Further Information
Get more info on 'Figure-eight Knot Mathematics'.
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