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This volume contains papers based on recent research on quantum field theory, statistical mechanics, quantum groups and topology. ... Read more

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Recent work by mathematicians and physicists has uncovered revelatory connections between knot theory and the problem of developing a quantum theory of gravity.This book, the proceedings of a workshop held to bring together researchers in knot theory and quantum gravity, features a number of expository and research papers that will aid significantly in closing the gap between the two disciplines. It will serve as a guide for mathematicians and physicists seeking to understand this rapidly developing area of research.The book represents a state-of-the-art study of current research and progress.The editor is the author of Gauge Fields, Knots, and Gravity (World Scientific), a graduate level text on the topic. ... Read more

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Knot theory has recently emerged as a productive field of study in both the physical and mathematical sciences. This book is concerned with the fundamental question of the classification of knots, and more generally the classification of arbitrary (compact) topological objects which can occur in the normal space of physical reality.The author explains his classification algorithm--using the method of normal surfaces--in a simple and concise way. The reader is thus shown the relevance of such traditional mathematical objects as the Klein bottle or the hyperbolic plane to this basic classification theory.The Classification of Knots and 3-Dimensional Spaces will be of interest to mathematicians, physicists, and other scientists who want to apply this algorithm to their research in knot theory. ... Read more

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This volume explains how knot theory and Feynman diagrams can be used to illuminate problems in quantum field theory.The author emphasizes how new discoveries in mathematics have inspired conventional calculational methods for perturbative quantum field theory to become more elegant and potentially more powerful methods. The material illustrates what may possibly be the most productive interface between mathematics and physics. As a result, it will be of interest to graduate students and researchers in theoretical and particle physics as well as mathematics. ... Read more

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Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.

The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.

This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics.

From the contents:

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Deals with an areaof research that lies at the crossroads of mathematics and physics. The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions. Professor Atiyah presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential readingfor all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area. ... Read more

Customer Reviews (3)

Quick overview of TQFT and knot invariants
This book is a very quick overview of what was known at the time (1989) about the connection between quantum field theory and knot theory. The subject of topological quantum field theories and their connection with knot invariants was at that time just beginning thanks to the work of Edward Witten on the Jones polynomial.

The approach that the author takes in the book is very formal and not for the beginner who is looking to learn about these results. Readers with enough background to read it will no doubt want to read more up-to-date treatments of the subject. The book however does give an indication of how Feynman path integrals are used to define the invariants. The use of these is not rigorous mathematics and this has not changed at the present day.

Good intro to topological quantum field theory
As the above review indicates, this is not an introduction to knot theory.Those interested should first read Colin Adams "The Knot Book" and Kauffman's "Knots and Physics". This instead introduces thereader to the subject of topological quantum field theory, and shows how aFeynmann Path integral approach gives rise to the Jones polynomial knotinvariant (Witten's approach).

The first two chapters are accessible tothose who have had graduate-level abstract algebra and some topology. After that, a good familiarity with quantum field theory and quantizationof symplectic manifolds (although not strictly speaking necessary) makesthe subjects clearer.

Those who are working in Donaldson andSeiberg-Witten approaches to four-dimensional topology will want to readthis book.

NOT a book about KNOTS!
If you are seeking a graduate level discussion on quantum theory and knot theory, this is your book. Definitely not for the amateur user... ... Read more

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The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed. ... Read more

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This volume is a collection of research papers devoted to the study of relationships between knot theory and the foundations of mathematics, physics, chemistry, biology and psychology. Included are reprints of the work of Lord Kelvin (Sir William Thomson) on the 19th-century theory of vortex atoms, reprints of modern papers on knotted flux in physics and in fluid dynamics, and knotted wormholes in general relativity. It also includes papers on Witten's approach to knots via quantum field theory and applications of this approach to quantum gravity and the Ising model in three dimensions. Other papers discuss the topology of RNA folding in relation to invariants of graphs and Vassiliev invariants, the entanglement structures of polymers, the synthesis of molecular Mobius strips and knotted molecules. The book begins with an article on the applications of knot theory to the foundations of mathematics and ends with an article on topology and visual perception. This volume should be of interest to workers interested in new possibilities in the uses of knots and knot theory. ... Read more

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A formal theory of why some crises end in war
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This volume provides an introduction to knot and link invariants as generalized amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally includes a range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part 1 is a systematic course on knots and physics starting from the ground up; and Part 2 is a set of lectures on various topics related to Part 1. Part 2 includes topics such as frictional properties of knots, relations with combinatorics and knots in dynamical systems. In this third edition, a paper by the author entitled "Knot Theory and Functional Integration" has been added. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral.While the book supplies the background, this paper can be read independently as an introduction to quantum field theory and knot invariants and their relation to quantum gravity. As in the second edition, there is a selection of papers by the author at the end of the book. Numerous clarifying remarks have been added to the text. ... Read more

Customer Reviews (1)

math grad.
Overview of knots/physics. The book is fairly self-contained. It also has lots of pictures and works through the mathematics.
Introduces bracket polynomial, temperly-lieb algebra, and modeling physics ideas out of this stuff. ... Read more

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This volume consists primarily of survey papers that evolved from the lectures given in the school portion of the meeting and selected papers from the conference. Knot theory is a very special topological subject: the classification of embeddings of a circle or collection of circles into three-dimensional space. This is a classical topological problem and a special case of the general placement problem: Understanding the embeddings of a space X in another space Y. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 25 years. From the Jones, Homflypt and Kauffman polynomials, quantum invariants of 3-manifolds, through, Vassiliev invariants, topological quantum field theories, to relations with gauge theory type invariants in 4-dimensional topology. More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study.These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book. It is a remarkable fact that knot theory, while very special in its problematic form, involves ideas and techniques that involve and inform much of mathematics and theoretical physics. The subject has significant applications and relations with biology, physics, combinatorics, algebra and the theory of computation. The summer school on which this book is based contained excellent lectures on the many aspects of applications of knot theory. This book gives an in-depth survey of the state of the art of present day knot theory and its applications. ... Read more

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Chapters: Figure-Eight Knot, Unknot, Knot Invariant, Braid Group, Ménage Problem, Crossing Number, Borromean Rings, Alexander Polynomial, Connected Sum, Solomon's Knot, History of Knot Theory, Linking Number, Skein Relation, Fox N-Coloring, Knots and Graphs, Jones Polynomial, Link Group, Satellite Knot, Khovanov Homology, Seifert Surface, Tangle, Eilenberg-mazur Swindle, Racks and Quandles, Brunnian Link, Chirality, Clasper, Chiral Knot, Planar Algebra, Braid Theory, Reidemeister Move, Signature of a Knot, Tricolorability, Alternating Knot, Tait Conjectures, Finite Type Invariant, Biquandle, Invertible Knot, Unknotting Problem, Torus Knot, Milnor Map, Ropelength, Trefoil Knot, Arf Invariant of a Knot, Conway Notation, Virtual Knot, List of Knot Theory Topics, Homfly Polynomial, Pretzel Link, Arithmetic Topology, Dowker Notation, Knot Polynomial, Link Concordance, Gordon-luecke Theorem, Fary-milnor Theorem, Average Crossing Number, Slice Genus, Stick Number, Knot Group, Prime Knot, Fibered Knot, Kauffman Polynomial, Writhe, Framed Knot, Knot Thickness, Temperley-lieb Algebra, Bracket Polynomial, Hyperbolic Link, Slice Knot, Hopf Link, Unlink, Whitehead Link, 2-Bridge Knot, Double Torus Knot, Crosscap Number, Möbius Energy, Mutation, Free Loop, Knot Complement, Birman-wenzl Algebra, Berge Conjecture, Ribbon Knot, Cinquefoil Knot, Volume Conjecture, Flype, Knot Tabulation, Kontsevich Invariant, Bridge Number, Hyperbolic Volume, Self-Linking Number, Regular Isotopy, Perko Pair, Milnor Conjecture, Band Sum, (-2, 3, 7) Pretzel Knot, Thurston-Bennequin Number, List of Mathematical Knots and Links, Split Link, Wild Knot, Berge Knot, Physical Knot Theory, Legendrian Knot, Alexander Matrix, Algebraic Link, Closed Braids, String Link, Milnor Invariants, Cable Knot. Source: Wikipedia. Pages: 361. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more th...More: http://booksllc.net/?id=153008 ... Read more

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Chapters: Tesseract, Fundamental Group, 3-Sphere, Polychoron, Borsuk-ulam Theorem, Euler Characteristic, Genus, Knot Theory, Homological Algebra, Winding Number, Sheaf, Covering Space, Homology Theory, Spectral Sequence, Cobordism, Abstract Polytope, Hopf Fibration, Vector Bundle, Fiber Bundle, Formal Group, Spin Structure, Surgery Theory, Algebraic K-Theory, Seifert-van Kampen Theorem, Cw Complex, Esquisse D'un Programme, List of Cohomology Theories, Lefschetz Hyperplane Theorem, Intersection Homology, Simplicial Set, Nonholonomic System, Twisted K-Theory, Higher-Dimensional Algebra, Associated Bundle, Line Bundle, Massey Product, Real Projective Space, Barycentric Subdivision, Simply Connected Space, Vietoris-rips Complex, Triangulation, Characteristic Class, Čech Cohomology, Künneth Theorem, Clique Complex, Steenrod Algebra, Simplicial Complex, Abstract Simplicial Complex, Duoprism, Acyclic Model, Complex Cobordism, Simplicial Homology, Monodromy, Betti Number, Delta Set, Sheaf Cohomology, Classifying Space, Lefschetz Fixed-Point Theorem, Free Product, Cup Product, Degree of a Continuous Mapping, Eilenberg−maclane Space, Hurewicz Theorem, Combinatorial Map, Cellular Approximation, Solenoid, Serre-swan Theorem, Calculus of Functors, Induced Homomorphism, Topological Modular Forms, Ramification, L-Theory, Surgery Structure Set, Riemann-hurwitz Formula, Mapping Cone, Mapping Cylinder, Topological K-Theory, Invariance of Domain, Semi-Locally Simply Connected, A∞-Operad, Duocylinder, Aspherical Space, Crossed Module, Rose, Thom Space, Lusternik-schnirelmann Category, Homotopy Lifting Property, E∞-Operad, Serre Spectral Sequence, Cap Product, Semi-S-Cobordism, List of Algebraic Topology Topics, Alexander-spanier Cohomology, Microbundle, Eilenberg-zilber Theorem, Alexander Duality, R-Algebroid, Chain, Gysin Sequence, Morava K-Theory, Virtual Knot, Fundamental Class, Lie Algebra Bundle, Homotopy Extension Property, Universal Coef...More: http://booksllc.net/?id=245466 ... Read more

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This book was originally prepared as notes for an art/mathematics conference, and later used for other lectures and workshops on harmonics. It includes the Lambdoma table of frequencies and wavelengths to be used to determine the musical harmonics of lecture rooms, as well as other data compiled. It became the author's reference manual with which to make analogies between space/time, symbols, and other phenomena which could be translated into musical harmonics. (From the Preface to the Second Edition) ... Read more