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In February 2001, scientists at the Department of Energy's Los Alamos National Laboratory announced that they had recorded a simple knot untying itself. Crafted from a chain of nickel-plated steel balls connected by thin metal rods, the three-crossing knot stretched, wiggled, and bent its way out of its predicament--a neat trick worthy of an inorganic Houdini, but more than that, a critical discovery in how granular and filamentary materials such as strands of DNA and polymers entangle and enfold themselves.
A knot seems a simple, everyday thing, at least to anyone who wears laced shoes or uses a corded telephone. In the mathematical discipline known as topology, however, knots are anything but simple: at 16 crossings of a "closed curve in space that does not intersect itself anywhere," a knot can take one of 1,388,705 permutations, and more are possible. All this thrills mathematics professor Colin Adams, whose primer offers an engaging if challenging introduction to the mysterious, often unproven, but, he suggests, ultimately knowable nature of knots of all kinds--whether nontrivial, satellite, torus, cable, or hyperbolic. As perhaps befits its subject, Adams's prose is sometimes, well, tangled ("a knot is amphicheiral if it can be deformed through space to the knot obtained by changing every crossing in the projection of the knot to the opposite crossing"), but his book is great fun for puzzle and magic buffs, and a useful reference for students of knot theory and other aspects of higher mathematics. --Gregory McNamee
Book Description
Over a century old, knot theory is today one of the most active areas of modern mathematics. The study of knots has led to important applications in DNA research and the synthesis of new molecules, and has had a significant impact on statistical mechanics and quantum field theory. Colin Adams's The Knot Book is the first book to make cutting-edge research in knot theory accessible to a non-specialist audience. Starting with the simplest knots, Adams guides readers through increasingly more intricate twists and turns of knot theory, exploring problems and theorems mathematicians can now solve, as well as those that remain open. He also explores how knot theory is providing important insights in biology, chemistry, physics, and other fields. The new paperback edition has been updated to include the latest research results, and includes hundreds of illustrations of knots, as well as worked examples, exercises and problems. With a simple piece of string, an elementary mathematical background, and The Knot Book, anyone can start learning about some of the most advanced ideas in contemporary mathematics.
Customer Reviews:
Good Introduction to Knots.......2007-07-03
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!
Written for a non-mathematician but certainly enjoyable by mathematicians!.......2006-11-16
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.
Pretty good introduction.......2005-03-31
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.
Great introduction to knot theory.......2003-03-09
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.
Excellent motivation for knot theory.......2002-06-27
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.
Book Description
Rolfsen's beautiful book on knots and links can be read by anyone, from beginner to expert, who wants to learn about knot theory. Beginners find an inviting introduction to the elements of topology, emphasizing the tools needed for understanding knots, the fundamental group and van Kampen's theorem, for example, which are then applied to concrete problems, such as computing knot groups. For experts, Rolfsen explains advanced topics, such as the connections between knot theory and surgery and how they are useful to understanding three-manifolds.
Besides providing a guide to understanding knot theory, the book offers "practical" training. After reading it, you will be able to do many things: compute presentations of knot groups, Alexander polynomials, and other invariants; perform surgery on three-manifolds; and visualize knots and their complements. It is characterized by its hands-on approach and emphasis on a visual, geometric understanding.
Rolfsen offers invaluable insight and strikes a perfect balance between giving technical details and offering informal explanations. The illustrations are superb, and a wealth of examples are included.
Book Description
In most mathematics textbooks, the most exciting part of mathematics--the process of invention and discovery--is completely hidden from the reader. The aim of Knots and Surfaces is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you.
Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement.
Customer Reviews:
EXCELLENT INTRODUCTION TO UNUSUAL MATHEMATICAL TOPICS.......2005-12-09
This book (and the author's companion book GROUPS AND SYMMETRY) are both worth getting. This one introduces Graph Theory, Surface Topology and Knots while the other one introduces Groups, Border Patterns and Wallpaper Patterns. Both books provided a guided approach through exercises, and both books have exceptional bibliographies, and suggestions for further experimentation. (My inability to visualize or deal with Knots is not the author's fault, however.)
Intellectual Treat.......2002-05-22
This is such a wonderful book. If you are interested in mathematics but aren't a mathematician this is the book for you. While reading it and working through the problems I really had the feeling that I was doing real mathematics vs just walking the dog type problems. I think this book is just as good if not better in some regards to Jeffery Weeks popular and excellent book The Shape of Space. After reading this book you will really understand some Topology,Graph Theory, and Knot Theory.
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Knots and Feynman Diagrams
Dirk Kreimer
Manufacturer: Cambridge University Press
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This volume explains how knot theory and Feynman diagrams can be used to illuminate problems in quantum field theory. The author emphasizes how new discoveries in mathematics have inspired conventional calculational methods for perturbative quantum field theory to become more elegant and potentially more powerful methods. The material illustrates what may possibly be the most productive interface between mathematics and physics. As a result, it will be of interest to graduate students and researchers in theoretical and particle physics as well as mathematics.
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An Introduction to Knot Theory (Graduate Texts in Mathematics)
W.B.Raymond Lickorish
Manufacturer: Springer
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The Knot Book
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Knots and Links (AMS Chelsea Publishing)
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Knots and Links
ASIN: 038798254X |
Book Description
This volume is an introduction to mathematical Knot Theory; the theory of knots and links of simple closed curves in three-dimensional space. It consists of a selection of topics which graduate students have found to be a successful introduction to the field. Three distinct techniques are employed; Geometric Topology Manoeuvres, Combinatorics, and Algebraic Topology. Each topic is developed until significant results are achieved and chapters end with exercises and brief accounts of state-of-the-art research. What may reasonably be referred to as Knot Theory has expanded enormously over the last decade and while the author describes important discoveries throughout the twentienth century, the latest discoveries such as quantum invariants of 3-manifolds as well as generalisations and applications of the Jones polynomial are also included, presented in an easily understandable style. Thus this constitutes a comprehensive introduction to the field, presenting modern developments in the context of classical material. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory although explanations throughout the text are plentiful and well-done. Written by an internationally known expert in the field, this volume will appeal to graduate students, mathematicians and physicists with a mathematical background who wish to gain new insights in this area.
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Symmetry, Ornament and Modularity (Series on Knots and Everything)
Slavik Vlado Jablan
Manufacturer: World Scientific Publishing Company
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- Not exactly friendly for beginners (3.5 stars max)
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Knots and Links
Peter R. Cromwell
Manufacturer: Cambridge University Press
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Knot Theory
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Handbook of Knot Theory
ASIN: 0521548314 |
Book Description
Knot theory is the study of embeddings of circles in space. Peter Cromwell has written a textbook on knot theory designed for use in advanced undergraduate or beginning graduate-level courses. The exposition is detailed and careful yet engaging and full of motivation. Numerous examples and exercises serve to help students through the material, while an instructor's manual is available online.
Customer Reviews:
Not exactly friendly for beginners (3.5 stars max).......2007-09-27
I'm midway through this book as I write. But since I'm a beginner in the field I'm probably a better judge of the book's presentation of fundamentals than I will be of its more advanced topics anyway. In that capacity, I'm disappointed with this book.
Some pluses: Peter Cromwell's (PC's) stated aim is to make the topic accessible to advanced undergraduates without a prerequisite course in topology. Necessary results from that field are presented as "facts" in Chapter 2. (Nonetheless, a course in graph theory is a stated prerequisite, which judging by Harvard's current catalogue doesn't seem to be standard in US undergrad curricula.) The bibliography is quite extensive. A publisher's blurb somewhere trumpets the "hundreds" of diagrams in the book -- an accurate figure, thanks to the Appendix of 86 knot diagrams, a bit more than a third.
So why disappointed? Because, among other things, the diagrams in the main text aren't nearly enough. PC's style is to minimize the number of diagrams used, and to rely instead on abstract, formal mathematical descriptions as much as possible. The long recital of definitions and theorems from topology -- which, by hypothesis, are subjects in which his expected reader lacks background -- is a relative desert of diagrams. The description of companion and satellite knots is accompanied by an unlabeled diagram that leaves one confused as to which knot is which. The description of Seifert surfaces is so abstract I found it impossible to visualize even on repeated readings, before I consulted another text. The description of the construction of Seifert matrices is also a bit Delphic in its concision. And even if a diagram were too much to ask, would it really have stressed PC to include a sentence saying that a "meridian" wraps round the torus the short way and a "longitude" the long way, instead of leaving these non-intuitive defnitions implicit in equations?
The most striking example of PC's diagram minimalism in my reading so far is his discussion of "surgery equivalence" (Chapter 6 @114-118). The original 1998 paper by Bar-Natan, Fulman & Kauffman, written for pros, has 17 or 18 line drawing figures, which are labeled with +'s and -'s and other tags to help you understand the text. PC's treatment, written as an introductory text, has only 7 unlabeled figures. (In both paper and book, a figure may contain several diagrams.) PC's figures are mostly based on figures in the original paper, some have been re-drawn with shading (not needed), but all are stripped of labels and other aids. The basic situation of the proof, the construction of "tubes" between surfaces, is clearly shown in Fig. 1 in the paper, but there isn't any corresponding illustration in the book. (The closest thing to one comes in the next chapter, but PC doesn't give you any head's-up about where to find it.) I strongly recommend the original paper as an aid to understanding this technique; it's available for free online as I write this.
I couldn't have made much progress in the early stages of this book without help from Colin Adams's "The Knot Book", published by Freeman. The overlap in coverage between the two books isn't perfect, but it's substantial. Adams's book is copiously illustrated, diagrams are keyed well to the text, and the text descriptions are down-to-earth. OTOH, he describes everything in a non-rigorous "popular" style, so it's no substitute for the precision of PC's book. But reading Adams makes one realize how easy it might have been for PC to preface his rigorous descriptions with more vivid ones, to help the newcomer to the field.
The book also lacks solutions or even hints to exercises. Apparently the publisher promises them, but PC's own website disclaims that they will be available anytime soon, if ever. Also, many of the exercises say "show" and others say "prove", but the distinction, if any, is not clear in context; often you're asked to "show" certain things are true "for any knot", e.g. @Ex.3.10.5. Lack of solutions and hints is an automatic 1-point deduction; at least a half-point for the obscurity (giving PC the benefit of the doubt since I'm still in media res), for an upper bound of 3.5 stars.
To give credit where credit is due, PC very swiftly and graciously replied to an email inquiry from me about a point that I'd misunderstood. I very much appreciate that, and it says good things about the author. But it's not a workable solution for all points of difficulty for one reader, much less all of them. I hope that PC will be a bit more indulgent to beginners in a future edition of this book.
Finally, a wag of the finger to the publisher. When I bought this book in 2005, the cover price was $40; as of this review it's gone up 30%, if we ignore Amazon's discount. It's a handsome book, printed on expensive coated stock, a kind Cambridge also uses for textbooks with lots of color. But all the illustrations are black-and-white line drawings -- no need for such fancy paper at all. Had the publisher made a more sensible production choice, maybe the price for the paperback could have stayed at a more student-friendly level.
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Introduction to Knot Theory (Graduate Texts in Mathematics)
R. H. Crowell , and
R. H. Fox
Manufacturer: Springer
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ASIN: 0387902724 |
Average customer rating:
- A good reference/second book on knot theory
- Good intro to knot theory, with a lot of technical detail
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On Knots. (AM-115)
Louis H. Kauffman
Manufacturer: Princeton University Press
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ASIN: 0691084351 |
Customer Reviews:
A good reference/second book on knot theory.......2001-05-28
I don't feel that this book would be the best systematic introduction to the subject (say, for a course on knot theory). However, once someone has been introduced to knot theory(say, via a topology of manifolds class, more elementary book such as Adams, Livingston, or even a more advanced book such as Zieshang-Burde, Lickorish or Rolfsen), this book is an excellent reference.
The strength of this book is the "hands on" explinations given about many of the standard topics on knot theory (Alexander polynomial, Skein invariants, covering spaces, etc.) and I feel that the author does a great job on relating many of the combinatorial invariants to the topology of the knot complement. Many informative illustrations and examples are provided. This is one of the first references I look to when I need a refresher on a topic, or if I encounter something in classical knot theory that I am unfamiliar with.
Also, this book is just plain fun to read!
Of course, this book is from the mid 80's and therefore does not cover some of the more modern material.
Frankly, I've found that anything written by professor Kauffman to be well written and worth reading.
Good intro to knot theory, with a lot of technical detail.......1999-01-10
Starting out with the basics, Kauffman moves on quickly to more difficult concepts using advanced math. The book has a great section on knot tricks, and a nice table of knots.
Average customer rating:
- Fun, yet brief at times
- Excellent!
- Good for an introduction
- A very thorough volume for the serious student
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Knot Theory (Carus Mathematical Monographs)
Charles Livingston
Manufacturer: The Mathematical Association of America
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ASIN: 0883850273 |
Book Description
Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots, including one of mathematics’ most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject - the Conway, Jones and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology.
Customer Reviews:
Fun, yet brief at times.......2006-08-10
I really do enjoy this book - but picked it up as a means of teaching myself Knot Theory... as was the case with many of my text books in college, brevity (for the sake of publishing costs) makes some concepts more of a challenge to grasp. Overall, the illustrations are great, and if you do the exercizes, the material tends to flow more easliy. It seemed to me the book worked backwards a bit - first covering a subject, than introducing it comprehensively later on - not what I'm used to.
Keep in mind, I'm not a Mathematician, merely a graduate student of mathematics, who is interested in learning about this subject on my own.
Excellent!.......2002-01-11
Livingston does a good job on basic knot theory in this text. While Adams seems to jump around a bit in his book, Livingston keeps a nice flow to his work. The proofs require another text and a good background in algebra to understand, but the problems are wonderful for a deeper understanding of the material.
Good for an introduction.......2000-11-14
This book is an excellent introduction to knot theory for the serious, motivated undergraduate students, beginning graduate students,mathematicains in other disciplines, or mathematically oriented scientists who want to learn some knot theory.
Prequisites are a bare minimum: some linear algebra and a course in modern algebra should suffice, though a first geometrically oriented topology course (e. g., a course out of Armstrong, or Guillemin/Pollack) would be helpful.
Many different aspects of knot theory are touched on, including some of the polynomial invariants, knot groups, Alexander polynomial and related abelian invariants, as well as some of the more geometric invariants.
This book would serve as a nice complement to C. Adams "Knot Book" in that Livingston covers fewer topics, but goes into more mathematical detail. Livingston also includes many excellent exercises. Were an undergraduate to request that I do a reading course in knot theory with him/her, this would be one of the two books I'd use (Adam's book would be the other).
This book is intentionally written at a more elementary level than, say Kaufmann (On Knots), Rolfsen (Knots and Links), Lickorish (Introduction to Knot Theory) or Burde-Zieshcang (Knots), and would be a good "stepping stone" to these classics.
A very thorough volume for the serious student.......2000-05-31
Livingston's book is very concise and dense. It contains a lot of information, but is not the kind of book you could sit down and read through from cover to cover. It is excellent as a reference, a sort-of knot theory encyclopedia.
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