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Author: Louis H Kauffman Publisher: World Scientific ISBN: 9814494097 Category : Mathematics Languages : en Pages : 788
Book Description
This invaluable book is an introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary 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 I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II 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.
Author: Louis H Kauffman Publisher: World Scientific ISBN: 9814494097 Category : Mathematics Languages : en Pages : 788
Book Description
This invaluable book is an introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary 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 I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II 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.
Author: Louis H Kauffman Publisher: World Scientific ISBN: 9814502375 Category : Mathematics Languages : en Pages : 739
Book Description
In this second edition, the following recent papers have been added: “Gauss Codes, Quantum Groups and Ribbon Hopf Algebras”, “Spin Networks, Topology and Discrete Physics”, “Link Polynomials and a Graphical Calculus” and “Knots Tangles and Electrical Networks”. An appendix with a discussion on invariants of embedded graphs and Vassiliev invariants has also been included.This book is an introduction to knot and link invariants as generalized amplitudes (vacuum-vacuum amplitudes) for a quasi-physical process. The demands of knot theory, coupled with a quantum statistical framework, create a context that naturally and powerfully includes an extraordinary 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 has the advantage of providing very direct access to the algebra and to the combinatorial topology, as well as the physical ideas. This book is divided into 2 parts: Part I of the book is a systematic course in knots and physics starting from the ground up. Part II is a set of lectures on various topics related to and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics and knots in dynamical systems.
Author: Louis H. Kauffman Publisher: World Scientific ISBN: 9814383007 Category : Mathematics Languages : en Pages : 865
Book Description
An introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics.
Author: Michael Francis Atiyah Publisher: Cambridge University Press ISBN: 9780521395540 Category : Mathematics Languages : en Pages : 112
Book Description
These notes deal with an area that lies at the crossroads of mathematics and physics and rest primarily on the pioneering work of Vaughan Jones and Edward Witten, who related polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions.
Author: Louis H. Kauffman Publisher: World Scientific ISBN: 9789810220044 Category : Science Languages : en Pages : 502
Book Description
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 will be of immense interest to all workers interested in new possibilities in the uses of knots and knot theory.
Author: Colin Adams Publisher: CRC Press ISBN: 100022242X Category : Mathematics Languages : en Pages : 1048
Book Description
"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject." – Ed Witten, Recipient of the Fields Medal "I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field." – Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deep mathematical connections, and critical applications. Many of the articles in this book are accessible to undergraduates who are working on research or taking an advanced undergraduate course in knot theory. More advanced articles will be useful to graduate students working on a related thesis topic, to researchers in another area of topology who are interested in current results in knot theory, and to scientists who study the topology and geometry of biopolymers. Features Provides material that is useful and accessible to undergraduates, postgraduates, and full-time researchers Topics discussed provide an excellent catalyst for students to explore meaningful research and gain confidence and commitment to pursuing advanced degrees Edited and contributed by top researchers in the field of knot theory
Author: Colin Conrad Adams Publisher: American Mathematical Soc. ISBN: 0821836781 Category : Mathematics Languages : en Pages : 330
Book Description
Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.
Author: Vladimir G. Turaev Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110434563 Category : Mathematics Languages : en Pages : 620
Book Description
Due to the strong appeal and wide use of this monograph, it is now available in its third 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. Contents: Invariants of graphs in Euclidean 3-space and of closed 3-manifolds Foundations of topological quantum field theory Three-dimensional topological quantum field theory Two-dimensional modular functors 6j-symbols Simplicial state sums on 3-manifolds Shadows of manifolds and state sums on shadows Constructions of modular categories
Author: Gerald H Thomas Publisher: World Scientific ISBN: 9814719943 Category : Mathematics Languages : en Pages : 853
Book Description
The first volume, Geometry, Language and Strategy, extended the concepts of Game Theory, replacing static equilibrium with a deterministic dynamic theory. The first volume opened up many applications that were only briefly touched on. To study the consequences of the deterministic approach in contrast to standard Bayesian approaches, the richness of applications, requires an engineering foundation and discipline, which this volume supplies. It provides a richer list of applications, such as the Prisoner's Dilemma, which extends the resonant behavior of Vol. 1 to more general time-dependent and transient behaviors.
Author: American Mathematical Society. Short Course Publisher: American Mathematical Soc. ISBN: 0821848283 Category : Mathematics Languages : en Pages : 360
Book Description
This volume is based on lectures delivered at the 2009 AMS Short Course on Quantum Computation and Quantum Information, held January 3-4, 2009, in Washington, D.C. Part I of this volume consists of two papers giving introductory surveys of many of the important topics in the newly emerging field of quantum computation and quantum information, i.e., quantum information science (QIS). The first paper discusses many of the fundamental concepts in QIS and ends with the curious and counter-intuitive phenomenon of entanglement concentration. The second gives an introductory survey of quantum error correction and fault tolerance, QIS's first line of defense against quantum decoherence. Part II consists of four papers illustrating how QIS research is currently contributing to the development of new research directions in mathematics. The first paper illustrates how differential geometry can be a fundamental research tool for the development of compilers for quantum computers. The second paper gives a survey of many of the connections between quantum topology and quantum computation. The last two papers give an overview of the new and emerging field of quantum knot theory, an interdisciplinary research field connecting quantum computation and knot theory. These two papers illustrate surprising connections with a number of other fields of mathematics. In the appendix, an introductory survey article is also provided for those readers unfamiliar with quantum mechanics.
Author: Louis H Kauffman
Author: Louis H Kauffman
Author: Louis H Kauffman
Author: Louis H. Kauffman
Author: Michael Francis Atiyah
Author: Louis H. Kauffman
Author: Colin Adams
Author: Colin Conrad Adams
Author: Vladimir G. Turaev
Author: Gerald H Thomas
Author: American Mathematical Society. Short Course