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Title: Knots : Mathematics with a Twist by Alexei Sossinsky, Giselle Weiss ISBN: 0-674-00944-4 Publisher: Harvard Univ Pr Pub. Date: 31 December, 2002 Format: Hardcover Volumes: 1 List Price(USD): $24.95 |
Average Customer Rating: 3 (3 reviews)
Rating: 3
Summary: It is not that bad, but full of mistakes
Comment: 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: 1
Summary: This book is *very* bad!
Comment: Don't buy this book if you are the least mathematically minded.
Well, unless you want a cheap laugh. (Or rather expensive, in
fact.)
Either something very sinister has happened during one of the
translations (russian->french->english), or I seriously doubt
mr. Sossinsky's ability to teach anyone anything 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 anyone with a mathematical education
had 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."
Sadly, the figure eight knot has Conway polynomial 1-x^2.
Scary that an "expert" in 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!
I will leave to the mathematician seeking entertainment to find
the rest of the (many) errors. Some are rather fun. For
instance, the author confuses a figure-eight knot with an unknot,
right after casually mentioning that his intuition of space is
"fairly well developed".
Let me conclude by saying that in the light of all the mistakes,
Sossinsky's "personal digression"-story about how he almost
dicovered Kauffman's construction of the Jones polynomial before
Kauffman did becomes quite amusing.
Let me also say that on the good side, the actual subjects
treated in the book are well chosen. (Except, the author
promises twice to get back to telling us about the Alexander
polynomial but he never does...) That may justify the one star
I've had to give the book. So, my advise is: read the contents
pages and go learn the theory from elsewhere.
Rating: 5
Summary: Untangling Mathematically
Comment: 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.
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