There is a rich interplay between knots and links on one side, and \(3\)-manifolds on the other. Not only knots and links produce examples of \(3\)-manifolds that are among the easiest to visualise, just by taking their exteriors, thanks to their combinatorial nature they also provide very convenient ways to encode manifolds, notably via Dehn surgery, but also via other constructions, like branched covers.
On the other hand, since knots are determined by their exteriors, according to C. McA. Gordon and J. Luecke’s fundamental result in [J. Am. Math. Soc. 2, No. 2, 371–415 (1989; Zbl 0678.57005)], they can be seen as \(3\)-manifolds and investigated as such. Following the seminal ideas of Thurston, the study of \(3\)-manifolds relies heavily on geometric structures in general, and hyperbolic geometry in particular. Techniques and results of hyperbolic geometry have thus been fruitfully applied to knot theory, as well. Indeed, although not all knots are hyperbolic (that is, their complements need not admit a complete hyperbolic structure), just as in the case of manifolds, a significant step in understanding knots is the study of hyperbolic ones. As a consequence of Mostow-Prasad’s rigidity theorem, geometric invariants of hyperbolic manifolds like the volume are in fact topological ones, and so provide invariants for hyperbolic knots as well.
Although this interaction between knots and hyperbolic geometry has permeated geometric topology over the last three to four decades, the one under review is the first book to focus precisely on both these topics at the same time. In this sense, it differs greately from other books, both on hyperbolic geometry and on knot theory.
In spite of what the title may suggest, Purcell’s book should be considered more like an introduction to hyperbolic geometry rather than to knot theory, where knots are central but only as an interesting and readily accessible source of examples. More precisely, the book addresses specifically the more combinatorial and computational aspects of hyperbolic geometry which, again, tend to be only touched upon in more traditional books on the subject.
The starting point is a well-known example described by Thurston. He showed that the complement of the figure-eight knot is hyperbolic by first finding an ideal triangulation of the manifold, then proving that the triangulation can be made geometric by using hyperbolic ideal tetrahedra, and finally verifying that the gluing results in a complete hyperbolic structure. The triangulation defined by Thurston can be read off from a diagram of the knot while the existence of a complete hyperbolic structure is ensured by finding solutions to a system of gluing and completeness equations. The same strategy can be applied to determine (explicit) hyperbolic structures for other manifolds with toric boundary, in particular for knots generalising the figure-eight. In increasing order of generality, the following are treated in detail in the book: double twist knots, two-bridge knots, and alternating knots. Of course, the examples provide opportunities to point out where and how the strategy may fail.
Although it is possible to establish hyperbolicity of a given manifold in non constructive ways, for instance by applying Thurston’s hyperbolization theorem for Haken manifolds, the approach considered in the book can be implemented to give computer-assisted proofs of hyperbolicity. The first example of software of this type is SnapPea, originally created by J. Weeks and subsequently further developed by several other mathematicians.
After a short introductory chapter providing basic terminology in knot theory and an overview of the context and problems that will be discussed, the core of the book is subdivided into three parts.
Part one (Chapters 1 to 6) begins with a detailed explanation of Thurston’s figure-eight example. Starting from a diagram of the knot one can build a decomposition of its complement into ideal polyhedra. The combinatorics of the diagram and that of the decomposition are strictly related. This somehow illustrates and motivates how the rest of the book will unfold. Useful tools and results can also be found in this part: standard facts of hyperbolic geometry in dimensions 2 and 3, the notion of geometric structure, geometric triangulations, as well as more advanced topics like the thick-thin decomposition and (hyperbolic) Dehn filling.
Part two (Chapters 7 to 12) deals with the different classes of examples mentioned above: explicit geometric polyhedral decompositions are constructed for twist knots, two-bridge knots and links, and alternating knots and links. In this part augmented links are also defined. These are links obtained by adding to a knot (or link) trivial knot components, each encircling different pairs of strands twisted together. Their interest comes from the fact that by performing Dehn surgery on some extra trivial component one can add an even number of positive or negative crossings to the encircled twisted strands. Most fully augmented links, that is augmented links where no string of two or more crossings is present, are hyperbolic and can be used to produce hyperbolic knots and links by hyperbolic Dehn filling. Chapter 9 is devoted to the notions of volume and angle structures for a triangulation. Finally, properties of essential surfaces in knot and link complements are also discussed here. The notion of normal surface with respect to a triangulation is heavily exploited in this part. As already mentioned, a central idea is that the combinatorics of a knot or link diagram determines a polyhedral decomposition of its complement and hence a triangulation. Properties of the diagram translate into properties of the triangulation and vice-versa. Passing from one point of view to the other is crucial in several arguments. Schematic figures are provided to illustrate different situations discussed but their interpretations may not be as limpid as probably intended.
Part three (Chapters 13 to 15) is shorter and dedicated to knot invariants derived from the hyperbolic structure, that is the volume, Ford domains and canonical polyhedra, and the \(A\)-polynomial, which is defined both in terms of the character variety of a knot and in terms of the gluing and completeness equations.
The author made a visible effort to make her book self-contained. It should be thus accessible to students having a basic knowledge of topology, algebraic topology, and differential geometry.
More experienced readers will appreciate the presence of alternative proofs to some classical results as well as several more recent results. Although these are sometimes proposed in weaker forms to make them more accessible and limit technicalities, they offer some insight on how the tools developed in the book can be exploited in more general settings.