Knots: Mathematics with a Twist [Englisch] [Taschenbuch]

Either something really disturbing has happened during one of the translations (russian->french->english), or I seriously doubt mr. Sossinsky's ability to teach anyone about knot theory.

Almost every single calculation in the book is wrong. Some of the errors are plain typo's, admitted. But others are so disturbingly wrong that I had to read the passages several times to believe that a mathematician could have written this.

One notable example is when the author calculates (correctly for once) the Conway polynomial of the trefoil knot to be 1+x^2. Then goes on (this is so good, I just have to quote it):

"A calculation similar to this one shows that the Conway polynomial for the figure eight knot (Figure 1.2) is equal to x^2+1: it is the same as that for the trefoil. The Conway polynomial does not distinguish the trefoil from the figure eight knot; it is not refined enough for that."

In fact, the figure eight knot has Conway polynomial 1-x^2. Scary that an expert on knot theory can make this error (three times in a row!). -Afterall, the simplest counterexample to whether the Conway polynomial is a perfect invariant is a very, very basic thing to know!

Other mistakes are rather amusing (even whilst still being annoying). For instance, the author confuses a figure-eight knot with an unknot, shortly after casually mentioning that his intuition of space is "fairly well developed".

Another thing that annoys me as a mathematician is the author's "personal digressions", trying to explain how the minds of mathematicians work and why mathematics can be beautiful in the same way as arts and music. The worst one of them is concerned with how the author *almost* discovered the Kaufmann construction of the Jones Polynomial before Kaufmann did. (At least, that's how it sounds to me.) In my opinion, either you try to explain some math, or you do pocket philosophy. -Not both at once!

On the good side, the actual subjects treated in the book are very well chosen. (Except, the author promises twice to get back to telling about the Alexander polynomial but he never does...) (And that last thing reminds me: The book has no index!!!)

So, my advise is: read the contents pages and go learn the theory from elsewhere.

On the other hand, I thought explanations were pretty good.

So I would certainly not recommend it as a starter, but if you know enough of knot theory, the mistakes should keep you entertained...

Another error is made when giving an example of calculating the Conway polynomial for a link with two separate circles (page 68): the right-hand side of the equation should have no term in x. Figure 2.15 (algebraic representation of a braid) also has an error: the upper-right-hand braid elementary braid is b2, not b1. (The text below the diagram is correct, but the diagram itself has it wrong.)

For a beginner who is learning the subject, the necessity of sorting out the author's errors is unacceptable. A book with so many errors should have an errata (list of corrections) on the web, but I searched and found none.

I though the braid chapter was well-written. I have not studied braids before and it made the situation pretty clear.

On the plus side, the drawings are excellent, the best I have seen in any knot book. For example, figure 3.3 (page 40) has a nice diagram clearly showing various "problems" that might happen momentarily during Reidemeister moves. In this case, a picture is worth a thousand words.

I did not enjoy the author's mini-digressions into non-mathematical applications of knots. They went on too long and didn't relate well to the mathematics in the book.

Finally, this author seems to have a bit of an attitude. He makes it sound like he almost beat Kaufmann to discovering Kaufmann's bracket. Then he goes on to point out that the Celtic people discovered a form of it centuries ago (beating Kaufmann). Sounds like sour grapes to me. He makes frequent comments such as "the attentive reader will notice," which I found annoying after a while. Readers do not like to be insulted.

After a full day with this book, I am tossing it into the trash. The Knot Book by Colin Adams is solid on the math and a better overall introduction to the math side.