33 of 34 people found the following review helpful:
5.0 out of 5 stars
Excellent motivation for knot theory, June 26, 2002
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.
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20 of 24 people found the following review helpful:
4.0 out of 5 stars
Intelligent and intriguing!, January 18, 2001
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
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