Book Description

Customer Reviews (1)

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

Amazon.com
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 McNameeBook 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. ... Read more

Customer Reviews (9)

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.

Great introduction to knot theory
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
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. ... Read more

Book 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.

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. ... Read more

Book 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 many mathematics courses at the undergraduate level such as liberal arts math, and topology. Additionally, the book could easily challenge high school students in math clubs or honors math courses and is perfect for the lay math enthusiast.

Book Description
High-dimensional knot theory is the study of the embeddings of n-dimensional manifolds in (n+2)-dimensional manifolds, generalizing the traditional study of knots in the case n=1. The main theme is the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory. Many results in the research literature are thus brought into a single framework, and new results are obtained. The treatment is particularly effective in dealing with open books, which are manifolds with codimension 2 submanifolds such that the complement fibres over a circle. The book concludes with an appendix by E. Winkelnkemper on the history of open books. ... Read more

Book Description
Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. The book begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories, as well as lattice techniques.The authors then discuss applications in quantum gravity in detail.Following chapters consider knot theories, braid theories and extended loop representations in quantum gravity.A final chapter assesses the current status of the theory and points out possible directions for future research.Download Description
Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. The book begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories, as well as lattice techniques.The authors then discuss applications in quantum gravity in detail.Following chapters consider knot theories, braid theories and extended loop representations in quantum gravity.A final chapter assesses the current status of the theory and points out possible directions for future research. ... Read more

Customer Reviews (1)

Good introduction to some of the early research
The search for a theory of quantum gravity has occupied the time of a large number of researchers for over half a century, but as yet there is hardly any agreement on the conceptual foundations of such a theory, and this situation is aggravated by the lack of any experimental results that would drive its construction. Such evidence is absolutely necessary, for the lack of it sometimes leads the researcher into making wild speculations, which even though they are interesting from a mathematical standpoint, cannot be distinguished from alternatives that also have no experimental foundation. This situation has not prevented researchers from trying to find a theory of quantum gravity, and some of them proceed by analogy to what is done in quantum field theories that are well understood, such as quantum electrodynamics. Other researchers, particularly those that have a sophisticated mathematical background, have chosen string theory as the best candidate for a quantum theory of gravity.

As is readily apparent in the forward to this book, the authors favor the first approach, believing that quantum gauge theories, of which quantum electrodynamics is a primary example, offer the best hope for guidance in constructing a viable quantum gravity. They emphasize though a very particular aspect of these theories, namely that the requirement for gauge invariance forces one to view the "Wilson loops" as being the entities of primary importance. But more importantly, the authors assert that that Wilson loops allow one to gain information in the non-perturbative realm of quantum field theory. Calculations in non-perturbative quantum field theory are notoriously difficult, even though some progress has been made in the area of lattice gauge theories, so any insight the authors can offer in this regard is of utmost importance. Hence this book should be viewed as a study of quantum observables on the loop space. The authors hope that these observables, called `Schwinger functions' in the perturbative realm, will along with the differential equations and boundary conditions that determine them, will give a viable theory of quantum gravity.

The differential geometry of gauge theories is usually done using the formalism of principal fiber bundles. Classical gauge fields are viewed as sections of these bundles, and the results of non-trivial field interactions are compared from point to point by the use of parallel transport along curves defined in the base spaces of these bundles. This comparison is done with a `connection' on the bundle, and for a closed curve the failure of an entity to return to its original value after traversing the curve is taken to be a sign of non-trivial interactions or "curvature". Principal fiber bundles of course have an associated Lie group and elements of this group act on objects to parallel transport them along the closed curves. These group elements are thus dependent on the curve, and are called `holonomies'.

This is the classical picture, but what happens to this scenario in the quantum realm, and in this realm is it plausible to view it as a theory of quantum gravity?The authors spend the first six chapters discussing the loop group, and its use in the quantization of classical electrodynamics and classical Yang-Mills theory, as compared with what is done in the usual Hamiltonian formalism. The `quantum loop representation' plays a central role in their exposition, which is motivated by essentially two different approaches, one of which is essentially a Fourier transform of wavefunctions of the connection, while the other involves the quantization of a non-canonical algebra. In both cases the quantization procedure involves coming to grips with a constrained system, which as is well known is very challenging and the loop representation cannot be expected to be a panacea in this regard.

The trick involves the identification of the physical states taking into account diffeomorphism invariance and the Hamiltonian constraint. The authors do this for pure quantum gravity (no matter fields) using the Ashtekar formalism and `point-splitting' methods, reinforcing the idea of course that one is not going to escape the need for regularization, as is the case for all successful quantum field theories so far. The inclusion of matter fields is done for (uncharged) Weyl fermions, with the geometric interpretation that the Weyl part of the Hamiltonian is a translation operator in much the same way as the Hamiltonian in the case of pure gravity. The authors believe take this to mean that the loop representation for quantum gravity predicts the Dirac equation for fermions, but unfortunately they do not elaborate on this in much detail at all.

If loop quantization is to bring about a "unified" field theory in some sense then it must be able to show how the loops from one theory can be combined with the loops from another. The authors do this for the case of general relativity and electrodynamics, wherein a loop representation is introduced that is based on a single loop that accounts for the information of these two interacting theories. As expected, this involves enlarging the symmetry group SU(2) to U(2). They show that the wavefunctions for the unified loop representation depend on two loops, but that there is no effective distinction between the two loops. Generalizations to the case of Yang-Mills + general relativity are alluded to in the text but not discussed in any depth.


... Read more

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. ... 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

Book Description
Since discovery of the Jones polynomial, knot theory has enjoyed a virtual explosion of important results and now plays a significant role in modern mathematics. In a unique presentation with contents not found in any other monograph, Knot Theory describes, with full proofs, the main concepts and the latest investigations in the field.The book is divided into six thematic sections. The first part discusses "pre-Vassiliev" knot theory, from knot arithmetics through the Jones polynomial and the famous Kauffman-Murasugi theorem. The second part explores braid theory, including braids in different spaces and simple word recognition algorithms. A section devoted to the Vassiliev knot invariants follows, wherein the author proves that Vassiliev invariants are stronger than all polynomial invariants and introduces Bar-Natan's theory on Lie algebra respresentations and knots.The fourth part describes a new way, proposed by the author, to encode knots by d-diagrams. This method allows the encoding of topological objects by words in a finite alphabet. Part Five delves into virtual knot theory and virtualizations of knot and link invariants. This section includes the author's own important results regarding new invariants of virtual knots. The book concludes with an introduction to knots in 3-manifolds and Legendrian knots and links, including Chekanov's differential graded algebra (DGA) construction.Knot Theory is notable not only for its expert presentation of knot theory's state of the art but also for its accessibility. It is valuable as a professional reference and will serve equally well as a text 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

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. ... Read more

Customer Reviews (4)

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

Book Description
Covering an area of current intense research interest, this book is ideal for all mathematicians, undergraduate or at research level, as an introduction to the field. The book provides a thorough account of the mathematical theory of knots and its interactions with related fields. This new paperback edition places the book within reach of all individuals interested in this new and exciting field. ... Read more

Customer Reviews (1)

Unique approach; vibrant and captivating
This unique and vibrant book is an introductory book on knot theory that somehow sneaks in a lot more rigorous mathematics than you would expect.Without seeming overly difficult, it somehow manages to include a concise introduction to the basics of knot theory, basic topology, and even a little bit of graph theory, algebraic topology and the necessary algebra.

While assuming little background, this book covers an extraordinarily diverse range of material, and ties it all together.The book is written so that a third-year undergraduate could understand it, but it's interesting enough that a graduate student will still find it fascinating.

What I love most about this book is the choice and ordering of topics--the authors dive right into the material, going to some depth in exploring polynomial invariants before they even touch any "abstract nonsense" so to speak; the machinery is developed throughout the book, as it is needed, and as a result seems natural and fully motivated.

I think this book is excellent for self-study; it would also make a great textbook for a course, although to some extent the material in the course would be dictated by what the book covers.I also think that someone teaching a topology or graph theorycourses should keep this book in mind and recommend it to any students inquiring about connections to knot theory. ... Read more

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. ... Read more

Book Description
Braid theory and knot theory are related via two famous resultsdue to Alexander and Markov. Alexander's theorem states that anyknot or link can be put into braid form. Markov's theorem givesnecessary and sufficient conditions to conclude that two braidsrepresent the same knot or link. Thus, one can use braid theoryto study knot theory and vice versa.

In this book, the author generalizes braid theory to dimensionfour. He develops the theory of surface braids and applies it tostudy surface links. In particular, the generalized Alexanderand Markov theorems in dimension four are given. This book isthe first to contain a complete proof of the generalized Markovtheorem.

Surface links are studied via the motion picture method, and someimportant techniques of this method are studied. For surfacebraids, various methods to describe them are introduced anddeveloped: the motion picture method, the chart description, thebraid monodromy, and the braid system. These tools arefundamental to understanding and computing invariants of surfacebraids and surface links.

Included is a table of knotted surfaces with a computation ofAlexander polynomials. Braid techniques are extended torepresent link homotopy classes. The book is geared toward awide audience, from graduate students to specialists. It wouldmake a suitable text for a graduate course and a valuableresource for researchers. ... Read more

Book Description
The present volume is an updated version of the book edited by C N Yang and M L Ge on the topics of braid groups and knot theory, which are related to statistical mechanics. This book is based on the 1989 volume but has new material included and new contributors. ... Read more

Customer Reviews (1)

A good reference
This book is a collection of articles that are representative of the many exciting developments in knot theory that were occurring at the time of publication. Many of these developments have been extended and generalized since then, such as the theory of Vassiliev invariants, and thus the articles could be viewed as an introduction to this research. Hence it could still serve as a reference to the mathematical theory of knots and it relation to physics, via statistical mechanics and quantum field theory. I did not read all of the articles, so only a few of the ones I did will be reviewed here.

The article by Vaughan Jones on polynomial invariants for knots via von Neumann algebras begins the collection and was definitely the tone-setting one of the time, due to the new invariants of knots discovered by Jones. The article discusses how to construct a polynomial invariant for tame oriented links using certain representations of the braid group. By using Markov's theorem and a trace on a type II(1) von Neumann algebra, the author shows that the invariant depends only on the closed braid. The von Neumann algebra is generated by an identity and a collection of projections, which satisfy certain types of relations. These relations involve a complex parameter, and when this parameter satisfies certain conditions there exists a trace on the von Neumann algebra which in turn satisfy a collection of relations. The relations on the projections and the trace determine the structure of the von Neumann algebra up to *-isomorphism. That the projection relations are similar to Artin's presentation of the braid group was what Jones and others to develop invariants of links and knots based on this trace. In another article Jones then obtains a polynomial invariant in two variables for oriented links that uses a trace on Hecke algebras "of type A", which was inspired by the connections with von Neumann algebras. His discussion in this article points out the need for a better understanding of the topological interpretation of these invariants. Pointing out that a more in-depth understanding of subfactors of finite index would assist in this topological interpretation, in a later article Jones outlines in more detail what is known for subfactors of finite index. The index, as defined by Jones, measures the size of a subfactor in a II(1) factor. In addition, Hans Wenzl discusses Hecke algebras of type A and subfactors, and shows how to compute the Jones index using AF algebras.

The most provocative article in the book, and one not rigorous from a mathematical standpoint, is the article by Edward Witten on the quantum field theory and the Jones polynomial. The connection between these two seemingly disparate fields caused great excitiment in both the physics and mathematics communities, in spite of the fact that these results are unjustified mathematically, due to their reliance on path integrals. Witten was motivated in this article to find a three-dimensional interpretation of the Jones polynomial, which he does so via Yang-Mills theory in three dimensions. However, the Yang-Mills theory which he uses is not the standard one, but instead is based on the purely topological Chern-Simons theory. Witten considers the quantum field theory defined by the Chern-Simons theory and uses its gauge fields to define gauge-invariant observables. Because of the side-constraint of general covariance, these observables are chosen to be Wilson lines, which are independent of the metric. In an oriented three manifold Witten then considers oriented and non-intersecting knots and assigns a representation to each knots. Using the Chern-Simons three form Witten computes the path integral of the Wilsonobservables, and then proposes that these quantities are 3-dimensional interpretations of the Jones invariant. Witten first proves that the Chern-Simon form gives a meaningful quantum theory, i.e. that it is free from anomalies, and he justifies this by reducing the Chern-Simons invariant to a ratio of determinants, and then showing the absolute value of this ratio is the Ray-Singer analytic torsion. Witten then considers the calculation of the phase of the ratio, and then via the canonical quantization of the theory, shows how to obtain the desired knot invariants. ... Read more

Book Description
This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting 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
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

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. ... 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