As a discipline, knot theory has two attractive features: on the one hand, many of its statements are easily accessible to outsiders; on the other, it draws on techniques from many different parts of mathematics. Capturing both these features is likely impossible for any knot theory textbook, and so authors are forced to make a choice as to their intended audience: what level of mathematical sophistication will they assume and what techniques will they emphasize? Burde and Zieschang have written a knot theory text aimed at people with at least some graduate-level background in topology. The techniques they emphasize are those which are rooted in combinatorial 3-manifold topology and combinatorial group theory. These choices make their text much less accessible than, say, Colin Adams’ justifiably popular text [A] or Rolfsen’s somewhat more advanced text [R]. However, Burde and Zieschang have written an essential reference for those wanting to do research in knot theory or 3-manifold theory.
As an example of the sort background required, I sample the following instances of non-trivial topology:
throughout the book, basic homology theory and the fundamental group are employed
on page 7, the authors prove that various forms of knot equivalence are equivalent; the proof rapidly and deftly applies triangulations.
Chapter 8 uses covering spaces to introduce and study the Alexander module and polynomial of a knot
Chapter 15 uses basic 3-manifold techniques (without providing many pictures) to study the relationship between the topology and algebra of knot complements.
If the reader either has the background or is capable of acquiring it rapidly, Burde and Zieschang’s text is an excellent reference for essential knot theory theorems and techniques. Particular strengths include providing a proof that the various forms of knot equivalence really are equivalent; a development of the topological and algebraic properties of 2-bridge knots; relationships between knots and braids; the definitions and properties of the Alexander, Jones, and HOMFLY-PT knot polynomials; and an explanation of the “Fox calculus”.
With the exception of a few difficult theorems (such as the infamous “Loop Theorem” from 3-manifold topology), the text is largely self-contained. It is certainly not easy reading, but the devoted reader will not find herself overly frustrated. There are numerous references for further reading, more detailed arguments, or original versions of proofs. A good companion text is [K] which covers a lot more ground, but includes far fewer proofs.
If a 3rd edition were produced, I would like to see more pictures and less use of Fraktur. For example, in one proof, both a Fraktur B and a Fraktur V are used within six lines of each other; the typesetting in the book makes them nearly indistinguishable. Those desires aside, I strongly recommend Burde and Zieschang’s text as a basic reference for anyone seeking knot theory knowledge.
[A]Colin Adams. The Knot Book. American Mathematical Society, 2004.
[K] Akio Kawauchi. A Survey of Knot Theory. Birkäuser, 1996.
[R] Dale Rolfsen. Knots and Links. American Mathematical Society, 2003.
Scott Taylor is a knot theorist and 3-manifold topologist at Colby College. Sadly, he has never figured out how to write in Fraktur.