Product 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. ... Read more

Customer Reviews (1)

Thorough and intriguing
The level of detail in this book is just right. It provides complete and rigorous proofs without getting bogged down in too much detail. In addition to some other knot theory standards, it has an excellent section on 3-manifold invariants arising from the Temperley-Lieb algebra. This book is appropriate for those who have a knowledge of algebraic topology. ... Read more

Product Description
Knot theory now plays a large role in modern mathematics. This unparalleled text and reference describes the main concepts of modern knot theory with full proofs accessible to both beginners and professionals alike. It presents both classical and modern knot theory, as well as the most significant results from braid theory, including the full proof of Markov's theorem, and Alexander's and Vogel's algorithms. It includes valuable information on the theory of coding knots by d- diagrams, as well as the author's own results in virtual knot theory. The material is presented at a level suitable for advanced undergraduate students, and the text is ideal for a course on knot theory. ... Read more

Customer Reviews (1)

Best of Knots
Knot Theory by Vassily Manturov (CRC Press) The aim of the present monograph is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. Thus, in the first chapter of the second part of the book (concerning braids) we start from the very beginning and in the same chapter construct the Jones two-variable polynomial and the faithful representation of the braid groups. A large part of the present title is devoted to rapidly developing areas of modern knot theory, such as virtual knot theory and Legendrian knot theory.
In the present book, we give both the "old" theory of knots, such as the fundamental group, Alexander's polynomials, the results of Dehn, Seifert, Burau, and Artin, and the newest investigations in this field due to Conway, Matveev, Jones, Kauffman, Vassiliev, Kontsevich, Bar-Natan and Birman. We also include the most significant results from braid theory, such as the full proof of Markov's theorem, Alexander's and Vogel's algorithms, Dehornoy algorithm for braid recognition, etc. We also describe various representations of braid groups, e.g., the famous Burau representation and the newest (1999-2000) faithful Krammer-Bigelow representation. Furthermore, we give a description of braid groups in different spaces and simple newest recognition algorithms for these groups. We also describe the construction of the Jones two-variable polynomial.
In addition, we pay attention to the theory of coding of knots by d-diagrams, described in the author's papers. Also, we give an introduction to virtual knot theory, proposed recently by Louis H. Kauffman. A great part of the book is devoted to the author's results in the theory of virtual knots.
Proofs of theorems involve some constructions from other theories, which have their own interest, i.e., quandle, product integral, Hecke algebras, connection theory and the Knizhnik-Zamolodchikov equation, Hopf algebras and quantum groups, Yang-Baxter equations, LD-systems, etc.
The contents of the book are not covered by existing monographs on knot theory; the present book has been taken a much of the author's Russian lecture notes book on the subject. The latter describes the lecture course that has been delivered by the author since 1999 for undergraduate students, graduate students, and professors of the Moscow State University.
The present monograph contains many new subjects (classical and modern), which are not represented in the author's earlier Russian version of this book.
While describing the skein polynomials we have added the Przytycky-Traczyk approach and Conway algebra. We have also added the complete knot invariant, the distributive grouppoid, also known as a quandle, and its generalisation. We have rewritten the virtual knot and link theory chapter. We have added some recent author's achievements on knots, braids, and virtual braids. We also describe the Khovanov categorification of the Jones polynomial, the Jones two-variable polynomial via Hecke algebras, the Krammer-Bigelow representation, etc.
The book is divided into thematic parts. The first part describes the state of "pre-Vassiliev" knot theory. It contains the simplest invariants and tricks with knots and braids, the fundamental group, the knot quandle, known skein polynomials, Kauffman's two-variable polynomial, some pretty properties of the Jones polynomial together with the famous Kauffman-Murasugi theorem and a knot table.
The second part discusses braid theory, including Alexander's and Vogel's algorithms, Dehornoy's algorithm, Markov's theorem, Yang-Baxter equations, Burau representation and the faithful Krammer-Bigelow representation. In addition, braids in different spaces are described, and simple word recognition algorithms for these groups are presented. We would like to point out that the first chapter of the second part (Chapter 8) is central to this part. This gives a representation of the braid theory in total: from various definitions of the braid group to the milestones in modern knot and braid theory, such as the Jones polynomial constructed via Hecke algebras and the faithfulness of the Krammer-Bigelow representation.
The third part is devoted to the Vassiliev knot invariants. We give all definitions, prove that Vassiliev invariants are stronger than all polynomial invariants, study structures of the chord diagram and Feynman diagram algebras, and finally present the full proof of the invariance for Kontsevich's integral. Here we also present a sketchy introduction to Bar-Natan's theory on Lie algebra representations and knots. We also give estimates of the dimension growth for the chord diagram algebra.
In the fourth part we describe a new way for encoding knots by d-diagrams proposed by the author. This way allows us to encode topological objects (such as knot, links, and chord diagrams) by words in a finite alphabet. Some applications of d-diagrams (the author's proof of the Kauffman-Murasugi theorem, chord diagram realisability recognition, etc.) are also described.
The fifth part contains virtual knot theory together with "virtualisations" of knot invariants. We describe Kauffman's results (basic definitions, foundation of the theory, Jones and Kauffman polynomials, quandles, finite-type invariants), and the work of Vershinin (virtual braids and their representation). We also included our own results concerning new invariants of virtual knots: those coming from the "virtual quandle", matrix formulae and invariant polynomials in one and several variables, generalisation of the Jones polynomials via curves in 2-surfaces, "long virtual link" invariants, and virtual braids.
The final part gives a sketchy introduction to two theories: knots in 3-manifolds (e.g., knots in RP3 with Drobotukhina's generalisation of the Jones polynomial), the introduction to Kirby's calculus and Witten's theory, and Legendrian knots and links after Fuchs and Tabachnikov. We recommend the newest book on 3-manifolds by Matveev.
At the end of the book, a list of unsolved problems in knot and link theory and the knot table are given.
The description of the mathematical material is sufficiently closed; the mono-graph is quite accessible for undergraduate students of younger courses, thus it can be used as a course book on knots. The book can also be useful for professionals because it contains the newest and the most significant scientific developments in knot theory. Some technical details of proofs, which are not used in the sequel, are either omitted or printed in small type. ... Read more

Product Description
Colin Adams, well-known for his advanced research in topology and knot theory, is the author of this exciting new book that brings his findings and his passion for the subject to a more general audience. This beautifully illustrated comic book is appropriate for undergraduates, high school students in math clubs or honors math courses and is perfect for the lay math enthusiast.The Tangle is the ultimate tool for knot theory because knots are defined in mathematics as being "closed on a loop". Readers use the Tangle to complete the experiments throughout the brief volume. ... Read more

Customer Reviews (1)

A complete introduction to an area of math that is neglected. That is unfortunate, as it is easily demonstrated
In my opinion, knot theory is an area of mathematics neglected in the elementary, middle and high schools. Knot theory offers the advantages that it can be visually displayed using inexpensive materials; each student can work alone or in groups and it is easily learned. After all, the Boy Scouts have been doing it for decades.
This book is a gentle introduction to the basic theory of knots, written at a level so that middle school students can understand it with a little help from the teacher. A cartoon format starring a math superheroine is used, so it is packed with easy to understand diagrams. The 62 pages of material is a complete introduction to the theory of knots and is also long enough so that a complete section could be offered. This book would also be suitable for teaching a lengthy session on using manipulatives in mathematics instruction offered to mathematics education majors.
... Read more

Product 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. ... Read more

Customer Reviews (5)

As good as it gets in introducing knot theory
As a survey of the basics of knot theory, this book is as good as it gets. The opening chapter is a history of knot theory, which is followed by a chapter on the mathematical definition of knots. The remainder of the book is a series of descriptions of knots, how they are represented, classified and the mathematical machinery used to transform them.
Very little in the way of deep mathematical knowledge is needed to understand the presentation, one of the most important requirements is the ability to think in spatial terms. Exercises are given at the end of each section although no solutions are provided.
Many areas of mathematics began as an abstract theory and after some time, applications are found. Knot theory is an element of this set; one of the applications is that it can be used to describe how proteins fold. A protein is a long chain of connected amino acids, but its' ability to be biochemically active is based on the structure that it folds into after construction. This book is a lively understandable introduction to this fascinating field; it is suitable for self-study or a special topics class in the area of knots.

Fun, yet brief at times
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!
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
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
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-ofknot theory encyclopedia. ... Read more

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Product 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.

Customer Reviews (7)

Basic introduction to knot theory within the grasp of the second-year undergraduate
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.

Ultimately: unsatisfying
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.

A Fun Book
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.

Read the Adams book instead
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.

It is not that bad, but full of mistakes
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... ... Read more

Product Description
Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. "The Knot Book" is an introduction to this rich theory, starting with our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. "The Knot Book" is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in "The Knot Book".Colin Adams received the Mathematical Association of America (MAA) Award for Distinguished Teaching and has been an MAA Polya Lecturer and a Sigma Xi Distinguished Lecturer. Other key books of interest available from the "AMS" are "Knots and Links" and "The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes".Amazon.com Review
In February 2001, scientists at the Department of Energy's Los AlamosNational Laboratory announced that they had recorded a simple knot untyingitself. Crafted from a chain of nickel-plated steel balls connected by thinmetal rods, the three-crossing knot stretched, wiggled, and bent its wayout of its predicament--a neat trick worthy of an inorganic Houdini, butmore 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 lacedshoes 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 cantake one of 1,388,705 permutations, and more are possible. All this thrillsmathematics professor Colin Adams, whose primer offers an engaging if challenging introduction to the mysterious, often unproven, but, hesuggests, ultimately knowable nature of knots of all kinds--whethernontrivial, satellite, torus, cable, or hyperbolic. As perhaps befits itssubject, Adams's prose is sometimes, well, tangled ("a knot is amphicheiralif it can be deformed through space to the knot obtained by changing everycrossing in the projection of the knot to the opposite crossing"), but hisbook is great fun for puzzle and magic buffs, and a useful reference forstudents of knot theory and other aspects of higher mathematics.--Gregory McNamee ... Read more

Customer Reviews (11)

..almost great
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.

bonne introduction
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.

Good Introduction to Knots
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!
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
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. ... Read more

Customer Reviews (2)

A good reference/second book on knot theory
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
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. ... Read more

Product Description
This book is based on a set of lectures presented by the author at the NSF-CBMS Regional Conference, Applications of Operator Algebras to Knot Theory and Mathematical Physics, held at the U.S. Naval Academy in Annapolis in June 1988. The audience consisted of low-dimensional topologists and operator algebraists, so the speaker attempted to make the material comprehensible to both groups. He provides an extensive introduction to the theory of von Neumann algebras and to knot theory and braid groups. The presentation follows the historical development of the theory of subfactors and the ensuing applications to knot theory, including full proofs of some of the major results. The author treats in detail the Homfly and Kauffman polynomials, introduces statistical mechanical methods on knot diagrams, and attempts an analogy with conformal field theory. Written by one of the foremost mathematicians of the day, this book will give readers an appreciation of the unexpected interconnections between different parts of mathematics and physics. ... Read more

Customer Reviews (1)

introduction to knot theory
This book asserts three subjects:

-subfactor theory: It gives the possible values of the index, nevertheless it is too sketchy and one solely interested in this subject should read his inventiones papers.

-knottheory: this is the best part of the books which explain withoutprerequisites and with great clarity the way to compute knotinvariants.

-a pot pourri about 'knot and statistical mechanics'generalization. This part is the worst since the book is now quite old andthe idea are not clear.

This book as the virtues of lectures (clarity,geometric ideas) and it's defaults (few computations, lack of rigor) ... Read more

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Customer Reviews (2)

text low quality
You can tell this is an older book by looking at the font and quality of the printing. It is low quality. I havn't looked into the book too much, so I can't comment on quality of the information, but it seems to be fairly standard in what it talks about. More of an upper lever introductory text. I would suggest looking into Livingston's book or Adams' as these are much more basic. However Livingston doens't include proofs as much and is kindof unclear on the geometrical approach. Still all are good books.

Brisk and intuitive way to dive into knot theory
Formal Knot Theory starts out with a planar diagram of a knot. Then the book shows how to label and name the intersections of the planar diagram. By the seventh page, you have an interesting group of terms and theorms. The author shows a method for converting the planar knot diagram into a Jordan curve with no crossing lines.

I am parked at Page 7 where the author introduces a Duality Conjecture. Here is where combinatorics and issues of Topology are introduced. A planar knot diagram has some very specific kinds of symmetry that can be observed using the labeling introduced. The symmetry and alternate ways of labeling a planar knot eventually develop into combinatorics and matrix statements.

The preparation required to work this book with ease is described as "advanced undergraduates and graduates".

I would ammend the "advanced undergraduate" qualification. This book has a reasonable price and very readable expansion of many knot ideas and problems. This is a good step beyond popular writing about math.This book is accessible and direct as distinguished from some other math books that are deliberately abstruse. Enough set and matrix math is modeled that I can follow along and consult other texts for examples and exersizes I need to work the examples.

Formal Knot Theory has the feel and pace of lecture material for a one semester course on Knots. For instance, I can't find in Formal Knot Theory a chapter heading associating knots with the Euler Polyhedron Theorm. But the Theorm is embedded in the ingenious labeling conventions introduced in the first chapter. ... Read more

Product 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. ... Read more

Customer Reviews (1)

Not exactly friendly for beginners (closer to 3 stars, actually)
The topics covered in this book are terrific. The presentation is disappointing. Before I'd finished the book I'd posted a review with 3.5 stars, now that I've completed a first reading of the book, I'd give it closer to 3. (Amazon won't let me revise the star count.)

Some pluses: In theory the book is 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, and is often relied on in the text.) The bibliography is quite extensive. A publisher's blurb somewhere trumpets the "hundreds" of diagrams in the book -- but more than a third of these appear in appendices and catalogues of knots. The discussion of arc presentations of braids in Ch. 10, a subject on which PC (the author) has published extensively, is quite interesting.

The main disappointment is that there aren't nearly enough diagrams in the main text, making many arguments hard to follow. PC relies instead on terse, formal mathematical descriptions as much as possible. Chapter 2's 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. Chapter 3's 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 in Chapter 6 is so abstract I found it impossible to visualize even on repeated readings, before I consulted another text. 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 (@10)? PC also often refers to diagrams in earlier chapters, thus chopping up your concentration by making you flip pages.

By contrast, compare any book by Kauffman (or even his original papers). They're very generous with diagrams, even incorporating them into lines of a proof. The original 1998 paper by Bar-Natan, Fulman & Kauffman, written for pros, is a much clearer exposition of the important concept of "surgery equivalence" than is PC's description for beginners (Chapter 6 @114-118). Even though most of PC's diagrams are based on the paper's, he uses only a few of them and has stripped them of helpful labels. (The paper is available for free online as I write this.)

Another sharp contrast is Colin Adams's "The Knot Book", published by Freeman.Although written more like a popularization than a math textbook, it has significant overlap with the book under review, even including some of the material on braids in PC's chapter 10. It made it relatively easy to grasp satellites/companions, the Seifert algortithm and many other topics, including the Kauffman bracket polynomial, another instance where PC is confusing despite his use of diagrams. I srongly recommend it as an adjunct read.

In addition to 1 point off for the obscure style, I automatically deducted 1 star because the book lacks solutions or even hints to exercises. The proofs of many significant theorems are left as exercises, so this is no small thing. (PC's own website disclaims that solutions 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.

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 everyone, or even for all the stuff that confused me. I hope that PC will be a bit more indulgent to beginners in a future edition of this book.

Finally, some wags of the finger to the publisher: (1) The blurb mentions applications of knot theory to chemistry, biology, etc., but such stuff occupies less than 1.5 pages out of 280+ of text (a biology example that is mentioned briefly, followed by cites to some papers (@212-213).) (2) When I bought this book in 2005, the cover price was $40; as of this review it's gone up 40+%, 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 reasonable and student-friendly level. ... Read more

Product 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. ... Read more

Customer Reviews (2)

EXCELLENT INTRODUCTION TO UNUSUAL MATHEMATICAL TOPICS
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
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. ... Read more

Product Description
Over the past 20-30 years, knot theory has rekindled its historic ties with biology, chemistry, and physics as a means of creating more sophisticated descriptions of the entanglements and properties of natural phenomena--from strings to organic compounds to DNA. This volume is based on the 2008 AMS Short Course, Applications of Knot Theory. The aim of the Short Course and this volume, while not covering all aspects of applied knot theory, is to provide the reader with a mathematical appetizer, in order to stimulate the mathematical appetite for further study of this exciting field. No prior knowledge of topology, biology, chemistry, or physics is assumed. In particular, the first three chapters of this volume introduce the reader to knot theory (by Colin Adams), topological chirality and molecular symmetry (by Erica Flapan), and DNA topology (by Dorothy Buck). The second half of this volume is focused on three particular applications of knot theory. Louis Kauffman discusses applications of knot theory to physics, Nadrian Seeman discusses how topology is used in DNA nanotechnology, and Jonathan Simon discusses the statistical and energetic properties of knots and their relation to molecular biology. ... Read more

Product Description

Knot theory is a concept in algebraic topology that has found applications to a variety of mathematical problems as well as to problems in computer science, biological and medical research, and mathematical physics. This book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fields.

The book contains most of the fundamental classical facts about the theory, such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials; also included are key newer developments and special topics such as chord diagrams and covering spaces. The work introduces the fascinating study of knots and provides insight into applications to such studies as DNA research and graph theory. In addition, each chapter includes a supplement that consists of interesting historical as well as mathematical comments.

The author clearly outlines what is known and what is not known about knots. He has been careful to avoid advanced mathematical terminology or intricate techniques in algebraic topology or group theory. There are numerous diagrams and exercises relating the material. The study of Jones polynomials and the Vassiliev invariants are closely examined.

"The book ...develops knot theory from an intuitive geometric-combinatorial point of view, avoiding completely more advanced concepts and techniques from algebraic topology...Thus the emphasis is on a lucid and intuitive exposition accessible to a broader audience... The book, written in a stimulating and original style, will serve as a first approach to this interesting field for readers with various backgrounds in mathematics, physics, etc. It is the first text developing recent topics as the Jones polynomial and Vassiliev invariants on a level accessible also for non-specialists in the field." -Zentralblatt Math

Product 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. ... Read more

Product Description
This is an introduction to the basic tools of mathematicsneeded to understand the relation between knot theory and quantumgravity. The book begins with a rapid course on manifolds anddifferential forms, emphasizing how these provide a proper languagefor formulating Maxwell's equations on arbitrary spacetimes. Theauthors then introduce vector bundles, connections and curvature inorder to generalize Maxwell theory to the Yang-Mills equations. Therelation of gauge theory to the newly discovered knot invariants suchas the Jones polynomial is sketched. Riemannian geometry is thenintroduced in order to describe Einstein's equations of generalrelativity and show how an attempt to quantize gravity leads tointeresting applications of knot theory. ... Read more

Customer Reviews (6)

Fantastic Text
I really enjoyed reading this book!A must have if you are interested in mathematical physics.Every page is a pedagogical masterpiece.

My favourite text of all time (so far)
This book should be at the top of anyone's reading list who is planning to get into serious mathematical physics.It deals with a good deal of complex material, but the presentation is easy to follow, and shouldn't be beyond most advanced undergraduates.There are a lot of good exercises which fill in most of the gaps. (If you want a book heavy on detail, this book may not be for you.If you want a book that gives you all the tools you're going to need to get start understanding quantum gravity and other areas in a short time, get this book immediately!)It's a shame the paperback edition doesn't seem to be available anymore; it's half the price, and checking with the publisher reveals that the paperback edition is still in print.

An excellent book !
Covers many topics in Mathematical Physics with great clarity. Highly recommended for those who are interested in a modern approach to Mathematical Physics.

Perfect
A beautifully written book which should be entitled "quantum gravity primer for the practical man".Clear and self-contained, this book will serve aa a small survey of mathematical physics, giving the readertools in particle physics and gravity.Excellently motivated topics. Compact enough to bring with you anywhere.The only thing it fails at isdicing a proper tomato.

Worth its weight in gold!
I think the review above by J. Pullin puts it very well. This is a great book, and a good place to get started (it also provides suggestions for further reading). The authors have done a fantastic job, and I highlyrecommend the book! ... Read more

Product Description
Those with an interest in knots, both young and old, will enjoy reading Why Knot? An Introduction to the Mathematical Theory of Knots.  Colin Adams, well-known for his advanced research in topology and knot theory, is the author of this new book that brings his findings and his passion for the subject to a more general audience.  Adams also presents a history of knot theory from its early role in chemistry to modern applications such as DNA research, dynamical systems, and fluid mechanics. Real math, unreal fun!

Each copy of Why Knot? is packaged with a plastic manipulative called the Tangle®. Adams uses the Tangle because “you can open it up, tie it in a knot and then close it up again.” The Tangle is the ultimate tool for knot theory because knots are defined in mathematics as being closed on a loop. Readers use the Tangle to complete the experiments throughout the brief volume. ... Read more

Product Description
This book is a survey of current topics in the mathematical theory of knots.For a mathematician, a knot is a closed loop in 3-dimensional space: imagine knotting an extension cord and then closing it up by inserting its plug into its outlet.Knot theory is of central importance in pure and applied mathematics, as it stands at a crossroads of topology, combinatorics, algebra, mathematical physics and biochemistry.

* Survey of mathematical knot theory
* Articles by leading world authorities
* Clear exposition, not over-technical
* Accessible to readers with undergraduate background in mathematics ... Read more

Product Description
The physical properties of knotted and linked configurations in space have long been of interest to mathematicians. More recently, these properties have become significant to biologists, physicists, and engineers among others. Their depth of importance and breadth of application are now widely appreciated and valuable progress continues to be made each year.This volume presents several contributions from researchers using computers to study problems that would otherwise be intractable. While computations have long been used to analyze problems, formulate conjectures, and search for special structures in knot theory, increased computational power has made them a staple in many facets of the field. The volume also includes contributions concentrating on models researchers use to understand knotting, linking, and entanglement in physical and biological systems. Topics include properties of knot invariants, knot tabulation, studies of hyperbolic structures, knot energies, the exploration of spaces of knots, knotted umbilical cords, studies of knots in DNA and proteins, and the structure of tight knots. Together, the chapters explore four major themes: physical knot theory, knot theory in the life sciences, computational knot theory, and geometric knot theory. ... Read more