The topics covered in this book are terrific. The presentation is disappointing. Before I'd finished the book I'd posted a review with 3.5 stars, now that I've completed a first reading of the book, I'd give it closer to 3. (Amazon won't let me revise the star count.)
Some pluses: In theory the book is accessible to advanced undergraduates without a prerequisite course in topology. Necessary results from that field are presented as "facts" in Chapter 2. (Nonetheless, a course in graph theory is a stated prerequisite, and is often relied on in the text.) The bibliography is quite extensive. A publisher's blurb somewhere trumpets the "hundreds" of diagrams in the book -- but more than a third of these appear in appendices and catalogues of knots. The discussion of arc presentations of braids in Ch. 10, a subject on which PC (the author) has published extensively, is quite interesting.
The main disappointment is that there aren't nearly enough diagrams in the main text, making many arguments hard to follow. PC relies instead on terse, formal mathematical descriptions as much as possible. Chapter 2's long recital of definitions and theorems from topology -- which, by hypothesis, are subjects in which his expected reader lacks background -- is a relative desert of diagrams. Chapter 3's description of companion and satellite knots is accompanied by an unlabeled diagram that leaves one confused as to which knot is which. The description of Seifert surfaces in Chapter 6 is so abstract I found it impossible to visualize even on repeated readings, before I consulted another text. And even if a diagram were too much to ask, would it really have stressed PC to include a sentence saying that a "meridian" wraps round the torus the short way and a "longitude" the long way, instead of leaving these non-intuitive defnitions implicit in equations (@10)? PC also often refers to diagrams in earlier chapters, thus chopping up your concentration by making you flip pages.
By contrast, compare any book by Kauffman (or even his original papers). They're very generous with diagrams, even incorporating them into lines of a proof. The original 1998 paper by Bar-Natan, Fulman & Kauffman, written for pros, is a much clearer exposition of the important concept of "surgery equivalence" than is PC's description for beginners (Chapter 6 @114-118). Even though most of PC's diagrams are based on the paper's, he uses only a few of them and has stripped them of helpful labels. (The paper is available for free online as I write this.)
Another sharp contrast is Colin Adams's "The Knot Book", published by Freeman. Although written more like a popularization than a math textbook, it has significant overlap with the book under review, even including some of the material on braids in PC's chapter 10. It made it relatively easy to grasp satellites/companions, the Seifert algortithm and many other topics, including the Kauffman bracket polynomial, another instance where PC is confusing despite his use of diagrams. I srongly recommend it as an adjunct read.
In addition to 1 point off for the obscure style, I automatically deducted 1 star because the book lacks solutions or even hints to exercises. The proofs of many significant theorems are left as exercises, so this is no small thing. (PC's own website disclaims that solutions will be available anytime soon, if ever.) Also, many of the exercises say "show" and others say "prove", but the distinction, if any, is not clear in context; often you're asked to "show" certain things are true "for any knot", e.g. @Ex.3.10.5.
To give credit where credit is due, PC very swiftly and graciously replied to an email inquiry from me about a point that I'd misunderstood. I very much appreciate that, and it says good things about the author. But it's not a workable solution for everyone, or even for all the stuff that confused me. I hope that PC will be a bit more indulgent to beginners in a future edition of this book.
Finally, some wags of the finger to the publisher: (1) The blurb mentions applications of knot theory to chemistry, biology, etc., but such stuff occupies less than 1.5 pages out of 280+ of text (a biology example that is mentioned briefly, followed by cites to some papers (@212-213).) (2) When I bought this book in 2005, the cover price was $40; as of this review it's gone up 40+%, if we ignore Amazon's discount. It's a handsome book, printed on expensive coated stock, a kind Cambridge also uses for textbooks with lots of color. But all the illustrations are black-and-white line drawings -- no need for such fancy paper at all. Had the publisher made a more sensible production choice, maybe the price for the paperback could have stayed at a more reasonable and student-friendly level.
