Knots and Primes. An Introduction to Arithmetic Topology
The theme of the present book is to build up an analogy between knot theory and number theory, which is baptized "arithmetic topology". The book starts with an informative introduction going back to the origins of both knots and arithmetic as is currently understood. It progresses to give a fairly complete account of the basic aspects of knots in three-manifolds before moving to arithmetic rings, number fields, and Galois groups.
In chapter 3, the book starts explaining the parallelisms between knot theory and number theory. where knots correspond to prime numbers. The bridge is the Galois group (Deck transformations) of the covering spaces on the topological side versus the Galois group of a ring or field extension on the arithmetic side. For instance, in chapter 4 it is shown an analogy between the linking number mod 2 of two knots and the Legendre symbol of two primes. This continues in the remaining 10 chapters, where many analogies between the two worlds are shown. The comparisons are slowly increasing in level of difficulty, getting to advanced and recent research in the last chapters. However, definitions are carefully formulated and proofs are largely self-contained throughout the book. When necessary, background information is provided and examples and illustrations are provided. Moreover, every chapter finishes with a conclusion of the objects which are analogous in both sides.
To read this book, it is convenient to have a good level of knowledge in number theory and in knot theory, at least at the graduate level. The analogy is most often used to go from the geometrical to the number theory side, but at some instances the direction is reversed (like in chapter 12 with the Iwasawa Main Conjecture). Most of the results mentioned in the second half of the book are of a very profound nature, and require some familiarity with both subjects.
What the book does not offer is any hint of the reason for the analogies. These are presented ad hoc. But why are there so many similarities? Should one expect the existence of a mechanism to go from knots to primes? Or in the other direction? Or maybe the parallelism is due to the fact that both theories can be expressed as examples of some meta-theory to be developed? (This is not completely outrageous since on the geometrical side the main object used is the fundamental group of the knot complement, which is an algebraic object.) These questions are not analysed or even mentioned. It seems to this reviewer that they are the main philosophical questions in the background.