Figure-eight knot (mathematics)

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 valueof 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,

: $F(x,\; y,\; z,\; t)=G(x,\; y,\; z^2-t^2,\; 2zt),$

where

: $G(x,y,z,t)=(z(x^2+y^2+z^2+t^2)+x(6x^2-2y^2-2z^2-2t^2),sqrt\{2\}tx+y(6x^2-2y^2-2z^2-2t^2)).$

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 have 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.

References

*Ian Agol, "Bounds on exceptional Dehn filling", Geometry and Topology 4 (2000), 431–449. MathSciNet|id=1799796
*Marc Lackenby, "Word hyperbolic Dehn surgery", Inventiones Mathematicae 140 (2000), no. 2, 243–282. MathSciNet|id=1756996
*Chun Cao and Robert Meyerhoff, "The orientable cusped hyperbolic 3-manifolds of minimum volume", Inventiones Mathematicae, 146 (2001), no. 3, 451–478. MathSciNet|id=1869847
*Robion Kirby, [http://math.berkeley.edu/~kirby/problems.ps.gz "Problems in low-dimensional topology"] , (see problem 1.77, due to Cameron Gordon, for exceptional slopes)
*William Thurston, [http://msri.org/publications/books/gt3m/ "The Geometry and Topology of Three-Manifolds"] , Princeton University lecture notes (1978-1981).