Knots : Mathematics with a Twist

Rating:
Summary: This book is bad!
Review: Don't buy this book if you're a mathematician!

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.

Rating:
Summary: It is not that bad, but full of mistakes
Review: I actually read the French version, and skimmed through the Englih one. When I read it in French, I was baffled by the number of mistakes per page. So I reread it, keeping a list of mathematical mistakes and typos(?). It averaged 1.7/page. I send it in to the French editor, but I realized that they kept the mistakes in the English version!

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...

Rating:
Summary: Read the Adams book instead
Review: If you just plan to skim the text and do not intend to try applying the ideas presented to actual knots, then you may not notice this small book's many errors. But if you wish to verify what the text says and try your hand at some knot calculations, then this is not the book for you. Perhaps the worst example is the author's comment that the figure-eight knot and the trefoil not have the same Conway polynomial. They don't. After an hour of calculating and recalculating, it is frustrating to discover that the author, not the reader, is the one in error. That kind of elementary error makes one question the author's basic competence and knowledge of the field.

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.

Rating:
Summary: A Fun Book
Review: If you like mathematics, even if you did not major in math, read this book. It is written for both the non-mathematician and the Ph.D. mathematician. For a more rigorous introduction, see Prasolov and Sossinsky, Knots, Links, Braids and 3-Manifolds.

Rating:
Summary: Untangling Mathematically
Review: It is always surprising and pleasing to find that mathematicians are busy in their ivory towers looking at non-numerical concepts and even using small subjects to turn out tomes that are impenetrable to us non-mathematicians. If you want to spend a little time learning how mathematicians think about the lowly subject of knots, there is now a little book with good illustrations and explanations that may go over the heads of most people, but nonetheless demonstrates the high degree of effort in this mathematical field. _Knots: Mathematics with a Twist_ (Harvard) by Alexei Sossinsky (who is a professor of mathematics at the University of Moscow; this work is translated by Giselle Weiss) demonstrates well the complexity of a field that might at first seem unpromising but actually has important relevance to the real world.

The diagrams here, and there are many of them, are a great help. You could make your knot cross over and under an infinite number of different ways. But how different, and how can you tell the difference between one knot and another? There is, according to Sossinsky, no algorithm that works in every case of classification, not even an algorithm that can be taught to a computer. This is true even though the attempts at classification, with graphic or symbolic notation which cannot be reproduced here, are quite complicated. So, being able to tell one knot from another is the as yet unattained Holy Grail of knot theory. Interestingly, if you tie a knot, however simple, into a string, you cannot tie another knot, however complicated, into the string so that one knot will, when it meets the other, untie the string. The proof of the impossibility of one knot canceling out another is nicely sketched here. The chapters here are written more-or-less independently of one another, so that if one stumps you, you can try the next with a clean slate. For needed relief, Sossinsky has put in digressions (and labeled some of them as such) which any reader ought to be able to enjoy, like the one about the slime eel that knots itself for defenses (left trefoil knot). Some of the coincidences between knots, algebra, quantum theory, and other disparate lines of thought are really quite lovely, and indicate once again that no one knows where research in pure mathematics may lead or how practical it may turn out to be.

Sossinsky has a witty style, and acknowledges how strange this mathematical world must be for visitors. At one point in demonstrating the procedure for composing a knot from primes, he parenthetically says of the task of making a rigorous definition of what he has described intuitively, "I will leave to the reader already corrupted by the study of mathematics the task." He is a genial guide to a strange land.