Title: The Sphere-Filling Curve Defined by the Figure-Eight Knot
Abstract: The biggest result in topology in the last fifty years is Perelman’s proof of Thurston’s Geometrization Conjecture, which claims that all $3$-manifolds admit a natural rigid geometric structure. This conjecture includes the million-dollar Poincare Conjecture, a prize that Perelman won but refused to accept.
I will explain the meaning of Thurston’s conjecture, though not its long and difficult proof. The thoughts that led Thurston to his conjecture are fascinating. The beginning for Thurston was a claim of Robert Riley that the complement of the figure-eight knot admits a geometric structure. Thurston had intuitive reasons for thinking that Riley’s result could not possibly be true. But eventually Thurston changed his mind and proceeded to show that almost all $3$-manifolds do in fact admit natural geometric structures, a result eventually subsumed by Perelman’s proof of the complete conjecture.
That Riley’s result was true led Thurston to another conjecture, less important but fascinating, that the figure-eight knot defines a natural $2$-sphere-filling curve (like the famous but artificial space-filling curves of Peano, Hilbert, and Polya).