Hyperbolic Knot Theory

Thurston's 1978 notes on The Geometry and Topology of Three-Manifolds have had a huge influence on at least a couple of generations of geometric topologists. They were passed around within the community and are now available online but were never officially published. In 1997, a book based on the notes appeared, titled Three-dimensional Geometry and Topology, Volume 1. This cleaned up and expanded on a number of topics from the notes, but covered only a subset of the material. Sadly, volume 2 never appeared. 

 

Purcell's Hyperbolic Knot Theory fills part of the gap here: the first few chapters give a clear and welcoming exposition of the material in chapters three and four of Thurston's notes. Later chapters cover many newer developments.

 

Thurston classified knots into torus knots (which can be drawn on the surface of a torus), satellite knots (each of which is contained in a tubular neighborhood of some simpler knot), and hyperbolic knots (whose complements admit complete hyperbolic metrics). Torus knots are relatively simple and well-understood, and satellite knots can be studied by looking at their constituent knots. From the perspective of this classification, hyperbolic knots are the least well-understood. Fortunately, the hyperbolic condition is very powerful: by results of Mostow and Prasad, the hyperbolic structure is unique. Together with a result of Gordon and Luecke, the geometry becomes a complete invariant of the knot. A full understanding of hyperbolic geometry on knot complements would then give a classification of hyperbolic knots.

 

Purcell’s book gives a comprehensive and up-to-date introduction to the resulting application of hyperbolic geometry to knot theory. Flowing in the other direction, knots provide a rich and accessible source for examples of spaces admitting hyperbolic structures. Many results and conjectures about general hyperbolic manifolds were first thought about in the context of knots. 

 

The first two parts of the book cover these two directions. The first develops the theory of hyperbolic structures on manifolds, taking knot complements as motivating examples. The second looks at various important classes of hyperbolic knots and introduces tools (some from general hyperbolic geometry) used to study them. The third and final part explores some important knot invariants coming from hyperbolic geometry.

 

This is a textbook aimed at an early graduate student level. Readers should have taken a course in basic topology, and some familiarity with Riemannian geometry will be helpful. Later on, it will also be useful to know about a few concepts from a first course in algebraic topology, for example the fundamental group. Many well-illustrated examples are given, as well as many useful exercises. The book is well referenced and indexed.

 

There are many existing books on hyperbolic geometry and on knot theory taken separately, but to my knowledge this is the first that substantially focuses on the two fields together. The combination benefits each of the constituents. This book will be useful both as an introduction and as a reference for those interested in either (or both!) topics.

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Henry Segerman is an associate professor in the department of mathematics at Oklahoma State University. His research interests are in three-dimensional geometry and topology, and in mathematical art and visualization.