Knots: Mathematics with a Twist by Alexei Sossinsky (PDF)

Description

Ornaments and icons, symbols of complexity or evil, aesthetically appealing and endlessly useful in everyday ways, knots are also the object of mathematical theory, used to unravel ideas about the topological nature of space. In recent years knot theory has been brought to bear on the study of equations describing weather systems, mathematical models used in physics, and even, with the realization that DNA sometimes is knotted, molecular biology.This book, written by a mathematician known for his own work on knot theory, is a clear, concise, and engaging introduction to this complicated subject. A guide to the basic ideas and applications of knot theory, Knots takes us from Lord Kelvin’s early―and mistaken―idea of using the knot to model the atom, almost a century and a half ago, to the central problem confronting knot theorists today: distinguishing among various knots, classifying them, and finding a straightforward and general way of determining whether two knots―treated as mathematical objects―are equal.Communicating the excitement of recent ferment in the field, as well as the joys and frustrations of his own work, Alexei Sossinsky reveals how analogy, speculation, coincidence, mistakes, hard work, aesthetics, and intuition figure far more than plain logic or magical inspiration in the process of discovery. His spirited, timely, and lavishly illustrated work shows us the pleasure of mathematics for its own sake as well as the surprising usefulness of its connections to real-world problems in the sciences. It will instruct and delight the expert, the amateur, and the curious alike.

User’s Reviews

Editorial Reviews: Review “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“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“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.”―Choice“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.”―Library Journal“Knots is a spirited, timely, and sound book by a mathematician who knows the technicalities but writes appealingly for the general reader.”―“It’s not often that a book can take you to the frontiers of mathematical research as pleasantly as Knots. Using nothing more complicated than a few simple diagrams, Alexei Sossinsky leads his readers on a gripping journey to the cutting edge of modern topology, with a hint at its deep connections with quantum physics.”―“Knots have fascinated humanity since the dawn of civilization and have permeated virtually all aspects of our lives, from engineering to sailing to knitting. Anyone interested in the beauty and mystery of knots will enjoy this amazing book.”― Review [A] thought-provoking analysis of why technology has failed to live up to its promises. –Daniel Goroff, Professor of the Practice of Mathematics, Harvard University About the Author Alexei Sossinsky is Professor of Mathematics, University of Moscow. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐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.

⭐This book was interesting, but left me unsatisfied. It was an extremely small book. As a result, it couldn’t ever tell much of anything because it was in too much of a hurry to get to the next topic. The author also claimed that the book should be easily understandable to anyone. As I was reading, I kept thinking, “How would someone without upper-level mathematical training possibly understand this section.”Ultimately, I would not recommend this book. But if the goal of this book is simply to whet the appetite and cause the reader to look deeper into the subject, then I believe that his mission was a success.

⭐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.

⭐excellent introduction to abstract subjectrecommended

⭐Author doesnt describe technical terms well which is far beyond to understand for a common reader .

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