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Knots: Mathematics with a Twist Paperback – 16 Jul 2004

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Product details

  • Paperback: 160 pages
  • Publisher: Harvard University Press; New Ed edition (16 July 2004)
  • Language: English
  • ISBN-10: 0674013816
  • ISBN-13: 978-0674013810
  • Product Dimensions: 1.3 x 12.7 x 17.8 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 1,277,994 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Product Description


Indeed, knots are trendy and also accessible to recreational mathematicians. A sophisticated high school student might enjoy working out the math in this book, while a full-fledged math student would find it a charming tour of knot theory's greatest hits...An enjoyable math book and highly recommended. -- Amy Crunvard Library Journal 20030201 The author describes knot theory by chronicling its history. Beginning with Lord Kelvin's ill-conceived idea of using knots as a model for the atom, Sossinsky moves to the connection of knots to braids and then on to the arithmetic of knots. Other topics are the Jones polynomial, which links knot theory to physics, and a clear exposition on Vassilev invariants. Throughout, this book untangles many a snag in the field of mathematics. Science News 20030111 In a charming and spirited discussion of classical and contemporary knot theory, Sossinsky, beginning with Lord Kelvin's (c. 1860) theory of knots as models for atoms...moves through discussions of braids, links, Reidemeister moves, surgery, various knot polynomials (Alexander-Conway, Homfly, Jones), Vassiliev invariants, and concludes with connections between and speculations about knots and physics. -- S. J. Colley Choice 20030901 This eminently likeable introduction to knot theory is heavily illustrated with diagrams to help us get our heads around the mind-bending ideas, and Sossinsky delights in breaking off at tangents to relate surprising knot-related facts of the natural world, such as the fish that ties its body in a knot to escape predators, or the topological operations that are performed by an enzyme on DNA. The Guardian 20040731

About the Author

Alexei Sossinsky is Professor of Mathematics, University of Moscow.

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Most Helpful Customer Reviews

3 of 3 people found the following review helpful By Eoin McAuley on 13 Sept. 2005
Format: Paperback
This is an excellent little book. If you've always wondered about knot theory, this provides all that you need to get you hooked on this fascinating topic.
The book explains everything in simple diagrams, and doesn't require any great proficiency in algebra. A geometrical intuition is more important. A number of topics are covered and they are treated independently so that you can dip into the book and just read a chapter without having read the rest.
The main topics covered are:
Knots as atoms. 2d representations of knots and Reidemeister moves. Braids. Invariants: the Conway polynomial; the Jones polynomial. The arithmetic of knots. Recent discoveries.

There are a number of minor mistakes in the book, from typographical errors (an x in the wrong place) to a wrong assertion (that the figure eight knot and the trefoil have the same Conway polynomial - they don't). These are not important and won't lead you too far astray if you are paying attention.
Nicely presented and nicely bound, this book is a delight!
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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 7 reviews
24 of 27 people found the following review helpful
This book is bad! 11 Aug. 2003
By Frank Nasser - Published on Amazon.com
Format: Hardcover
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.
13 of 13 people found the following review helpful
It is not that bad, but full of mistakes 22 Nov. 2003
By A Customer - Published on Amazon.com
Format: Hardcover
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...
17 of 21 people found the following review helpful
Read the Adams book instead 2 May 2004
By A Customer - Published on Amazon.com
Format: Hardcover
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.
5 of 6 people found the following review helpful
A Fun Book 23 Dec. 2004
By Michael Graber - Published on Amazon.com
Format: Paperback Verified Purchase
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.
0 of 1 people found the following review helpful
Basic introduction to knot theory within the grasp of the second-year undergraduate 26 Dec. 2007
By Charles Ashbacher - Published on Amazon.com
Format: Hardcover
Knot theory is one area of mathematics that has an enormous number of applications. The actual functionality of many biological molecules is derived largely by the way they twist and fold after they are created. Over the years, a great deal of mathematics has been invented to describe and compare knots.
The purpose of this book is to present the fundamentals of knot theory while avoiding the fine details as much as possible. Sossinsky has succeeded in doing that; he develops the machinery used to describe knots in a manner that is generally understandable. While it is necessary that the reader have some understanding of higher-level mathematics, the level does not have to be too high. It is well within the mathematical grasp of a second-year math undergraduate. There are many diagrams to aid in the understanding.
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