Hyperbolic Geometry on the Figure-Eight Knot Complement

The exact relationship between knot theory and non-euclidean geometry was a puzzle that survived more than 100 years. The histories of the two subjects were clearly intertwined; Carl Friedrich Gauss was a pioneer and did much to popularize both fields. After Gauss their two histories diverge a little, although taken together the list of luminaries in the two fields reads like a mathematical and scientific “Who’s Who?” of the 19th and early 20th century. Nikolai Lobachevsky, Felix Klein, and Henri Poincaré were three of the main developers of non-euclidean geometry while William Thomson (Lord Kelvin), James Clerk Maxwell, and Emil Artin were instrumental in fleshing out knot theory[4]. Although both subjects came to be recognized as following under the umbrella term “topology” and although giants such as Bernhard Riemann and Max Dehn [4, 7] worked in both areas the two fields remained disjoint until they were finally reunited in the 1970’s. In 1973 Robert Riley, then a graduate student at the University of Southampton in England, succeeded in showing that the figure-eight knot complement had a hyperbolic structure [4]. He did this by first showing that since fundamental group of the figure-eight knot complement is isomorphic to a subgroup of PSL2C, and then using the theory of Haken (of four color theorem fame) manifolds to show that the figure-eight knot complement is homeomorphic to H mod a discrete group of isometries [6]. Riley later showed that several other knot complements admit a hyperbolic structure and conjectured that indeed all knot complements except for torus and satellite knots admit a hyperbolic structure. It seems to often be the case in mathematics that the best way to make progress on a subject is to interest someone else in it. In 1977 Riley did exactly this when he met William Thurston at Princeton and motivated him to start investigating hyperbolic structures on knot complements [4]. Thurston soon came up with a more explicit way of showing that the figure-eight knot complement is hyperbolic. It is Thurston’s construction, which starts by gluing two tetrahedra together, that we will follow in Section 2. Relying in part on his experiences with knot complements [9] in 1978 Thurston completed his “hyperbolization theorem” or “geometrization theorem” (note that this is is a special case of the “geometrization