Customer Reviews

Knot Book

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4 star:		(3)
3 star:		(1)
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Knot theory has been a branch of mathematics that has been around for over a century, and now is finding applications in mnay areas, some of these being electrical circuit analysis, genetics, dynamical systems, and cryptography. This book, written for the layman or the beginning student of mathematics, is an excellent overview of what is known (and not known) in knot theory. Because of the pictorial nature of the subject, knot theory is an excellent way to get people interested in mathematics. Knot theory now is an established branch of mathematics, and it involves the use of tools from topology, analysis, and algebra. The problem of distinguishing one knot from another is one of the major questions in knot theory, and its partial resolution has been assisted by concepts from physics, namely statistical mechanics and quantum field theory. The author discusses the knot recognition problem, and other unsolved problems in the book, and he points out that in knot theory the unsolved problems can be approached by someone with very little background in advanced mathematical techniques. The author does an excellent job of introducing these problems and letting the reader experience, in his words, the joy of doing mathematics.

Chapter 1 is an introduction to the basic terminology of knot theory, and the author gives examples of the most popular elementary knots. He points out the historical origins of the theory, one of these being the attempt by Lord Kelvin to explain the origins of the elements, interestingly. The basic operations on knots are defined, such as composition and factoring, and the famous Reidemeister moves. The proof that planar isotopies and Reidemeister moves suffices to map one projection of a knot to another is omitted. After defining links and linking numbers, the author then discusses tricolorability, and uses this to prove that there are nontrivial knots.

Chapter 2 then overviews the strategies used in the tabulation of knots.The Dowker notation, used to describe a projection of a knot, is discussed as a tool for listing knots with 13 or less crossings. The author also discusses the Conway notation, and how it is used to study tangles and mutants. Graph theory is also introduced as a technique to study knot projections. The author discusses the unsolved problem of finding an elementary integer function that gives the prime knots with given crossing number, a problem that has important ramifications for cryptography (but the author does not discuss this application).

Since knots are complicated objects, then like many other areas in topology, the strategy is to assign a quantity to a knot that will distinguish it from all other knots. Such a quantity is called an invariant, and as one might guess, no one has yet found an invariant to distinguish all nontrivial knots from each other. In the last two decades though, new powerful knot invariants have been discovered, many of these being based on concepts from theoretical physics. In chapter 3, the author discusses the unknotting number, the bridge number, and the crossing number as elementary examples of knot invariants.

Chapter 4 is more complicated, in that the author shows how to use surfaces to assist in the understanding of knots. After discussing how to triangulate an surface and the concept of a homoeomorphism between surfaces, he introduces the Euler characteristic as an invariant of surfaces. Surfaces appear in knot theory as the space in the knot's complement, and the author introduces the concept of the compressibility of a surface, also very important in three-dimensional topology. Particular attention is paid to Seifert surfaces, which, given a particular knot, are orientable surfaces with one boundary component such that the boundary component is the knot in question.

Several different types of knots are considered in chapter 5, such as torus, satellite and hyperbolic knots. The latter are particularly interesting, since their study is part of the field of hyperbolic geometry, a subject that is now undergoing intense study. The author also introduces the theory of braids and the braid group. Not only are braids very important in the study of knots, but they have taken on major importance in cryptography and dynamical systems.

Chapter 6 is very interesting, and introduces some of the more contemporary topics in knot theory. The assignment of polynomials to knots goes back to the early 20th century, but it took the work of Vaughan Jones and his use of ideas from operator theory and statistical mechanics to provide polynomial invariants of knots that were much finer than the Alexander polynomial of the 1930s. The Jones polynomial however is not introduced the way Jones did, but instead via the Kaufmann bracket polynomial. The HOMFLY polynomial is introduced as a polynomial that generalizes the Jones and Alexander polynomials.

A few applications of knot theory are discussed in Chapter 7, such as the DNA molecule and topological stereoisomers. The author also discusses the applications of knot theory to the theory of exactly solvable models in statistical mechanics, a topic that has mushroomed in the past decade. This is followed by a brief overview of applications of knot theory to graph theory in chapter 8.

Chapters 9 and 10 are an introduction to knot theory as it relates to research in the topology of 3-dimensional manifolds and the existence of knots in dimensions higher than 3. The concepts introduced, particulary the idea of a Heegaard diagram, are used extensively in the study of 3-manifolds. In addition, the author mentions the famous Poincare conjecture, albeit in non-rigorous terms. The Kirby calculus, which is a kind of generalization of the Reidemeister moves, but instead models the sequence of operations that allow one to change from one Dehn surgery description of a 3-manifold to another is briefly discussed. The author also gives a few elementary, intuitive hints about how to visualize knotted objects in high dimensions.

Well-written, a good introduction to a mathematical research topic that requires only high-school level mathematics as background. Includes good applications to biology and chemistry, and written with a friendly, easy-going style.

I checked this book out of the library on the recommendation of a friend who was taking a knot theory class. While I am comfortable with calculus and differential equations, I have not had much experience with topology or group theory so I was hesitant. She assured me that I would understand the concepts presented there and that it would give a good introduction to the subject.

Wow! Was she ever right! First of all, the book is written in a clear and pleasant conversational style. The author does not hesitate to bring in examples or to show diagrams to clarify an idea. Indeed, with a subject such as knot theory, diagrams are essential! His use of exercises is well justified however, I would say that many laypersons are unfamiliar with proof techniques and thus might have some difficulties with several of those. Algebra is used sparingly at best as Adams prefers to let his words and images convey the ideas.

All in all, I would say that this book does a wonderful job of relating a subject which is at the forefront of mathematics, to the mathematically uninitiated. Hopefully, it will stimulate even further interest.

Owen

i've been studying knots independently for the better part of this month and using mr. adams's def. comes in handy. i have one major complaint that there are no answers for any of the exercises!!! i feel like some of the exercises he sets up are crucial proofs for understanding basic concepts, however they take several minutes to solve, and there is no follow-up to show if what you did was right or wrong! the result is just a lack of confidence in my understanding of the subject matter, and it just gets frustrating when the author does not address whether that knot's right, or whether that's not right.

This book is aimed at making knot theory accessible to people with little mathematical background, and it does so beautifully. However, the material is not watered down--and there is quite a lot of material in this book, as well as a number of open questions (which are quite difficult). The book starts with basics and seems easy, but it gets into challenging concepts rather quickly. Knot theory is one area of abstract mathematics that is particularly accessible to people with little background and this book works off this assumption quite well. Most importantly, this book is fun--it brings out the fun in the subject, and in mathematics in general!

This book would make excellent reading for anyone who likes puzzles, abstract thought, or novel forms of mathematics. It also would be interesting for mathematicians who want an introduction to knot theory. Someone who wants a more mathematical (but still accessible) treatment might want to check out "Knots and Surfaces" by N. D. Gilbert. In some respects it is a natural follow-up to this book. It is slightly more concise and has more rigorous mathematics in it.

Having first been exposed to interesting knots while in undergraduate courses in biology and chemistry and occasionally encountering knots in my mathematical life, I have long maintained a passing interest in the field. However, until now, no single event evoked a reaction strong enough to pique a desire to explore. All it took to change that was the reading of this book by Adams.
Surprisingly complete for an introductory text, it is also amazingly understandable. Requiring only knowledge of polynomials and a mind capable of understanding twists, I found it addictive. This is one area where it pays not to think straight. After reading it twice, I still pick it up and scan it in odd moments. Problems are scattered throughout the book, and many can be solved using only a piece of string. Those that are still unsolved are clearly marked, with is good, since the statements are often very simple.
There are many applications and the number is growing all the time. One of the most profound images and statements of discovery was the pictures of the knotting of the rings of Saturn and commentator Carl Sagan saying, "We don't understand that at all. We will have to invent a whole new branch of physics to understand it." The most esoteric recent explanation of the structure of the universe is the theory of superstrings, where all objects are multi-dimensional knots. A fascinating problem in molecular biology of the gene is the process whereby DNA coils when quiescent and uncoils to be copied. One chapter is devoted to applications, although more would have been helpful.
A non-convoluted introduction to the theory of convolutions, this book belongs in every mathematical library.

Published in Journal of Recreational Mathematics, reprinted with permission.

In terms of content, I would rate this book 4-5 stars. However, I rated it three stars because it had a flaw in terms of readibility. If you are willing to devote a lot of time to the subject, and are willing to take the time to work through all of the exercises, then this is the book for you. However, if you are just looking for some light reading on an unusual subject, there is a problem with the book. In many cases, if you don't complete the exercises, your ability to understand what follows in the chapter will be impaired. I bought the book to read on the train, and did not really have the facilities to work through all of the exercises. For me, the book would be greatly improved if solutions to some of the exercises (at least in sketch form) were included as an appendix.
In addition to being a good introduction to knots, the book also covers many othet topics in topology as well. At the end of the book, the author tries to show that there are practical applications to knot theory, but for the most part he appeared to be stretching. It seems that knot theory is pretty close to being "pure" mathematics. One thing that he did miss, however, was the application of knot theory to tying neckties. That would have been really practical!

I purchased this book as the textbook for a senior-level mathematics course in my undergrad. However, this book does not "read like a textbook." This book has been one of the easiest reads in a mathematics-heavy book I have ever had the pleasure of laying my eyes on. The pictures are clear, the words are concise, the ideas are organized logically and in proper order such that ideas are clearly described and explained in a manner that you don't have to be a mathematics major to understand what is being talked about.

While I purchased this book as a necessity for my coursework, if you have any kind of interest in mathematics (or just logic/puzzle games) the book is extremely interesting and you will learn A LOT.

Highly recommended. Very well-done book, especially for a topic as "obscure" as knot theory.

Ce livre est une bonne introduction à la théorie des naeuds: panorama impressionnant, il donne envie d'en savoir d'avantage. Je lui enlève une étoile car un certain nombre de dessin sont incompréhensibles.

One can make nothing wrong buying this book. It gives an easy introduction, and most parts are well explained. Don't expect to become an expert in knot theory after reading it but at least you are then familiar with the basics.