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 5/3,[2] and is achiral.

(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.

Many more have been discovered by generalizing Thurston's construction to other knots and links.

The figure eight knot complement is a double-cover of the Gieseking manifold, which has the smallest volume among non-compact hyperbolic 3-manifolds.

A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot.

A well-known conjecture is that the bound (except for the two knots mentioned) is 6.

in the Jones polynomial reflects the fact that the figure-eight knot is achiral.