Series on Knots & Everything - Böcker

This book is an introductory explication on the theme of knot and link invariants as generalized amplitudes (vacuum-vacuum amplitudes) for a quasi-physical process. The demands of the knot theory, coupled with a quantum statistical frame work create a context that naturally and powerfully includes an extraordinary range of interelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward the 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 with and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics, knots in dynamical systems.

This book is an introductory explication on the theme of knot and link invariants as generalized amplitudes (vacuum-vacuum amplitudes) for a quasi-physical process. The demands of the knot theory, coupled with a quantum statistical frame work create a context that naturally and powerfully includes an extraordinary range of interelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward the 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 with and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics, knots in dynamical systems.

This is a book of marvelous pictures that illustrates standard examples in low dimensional topology. The text starts at the most basic level (the intersection of coordinate planes) and gives hands on constructions of the most beautiful examples in topology: the projective plane, Poincare's example of a homology sphere, lens spaces, knotted surfaces, 2-sphere eversions, and higher dimensional manifolds. The text carefully explains the importance of the examples and the techniques without being bogged down in a morass of technicalities.Chapter 1 opens with the classification of orientable surfaces, and the meaning of space. Chapter 2 discusses examples of non-orientable surfaces including models of the projective plane and the Klein bottle. Chapter 3 discusses how curves fit on surfaces and gives a general discussion of knotted strings in space. In Chapter 4, some examples of other 3-dimensional spaces are described. These include the 3-dimensional sphere, lens spaces, and the quaternionic projective space. In Chapter 5, the author reviews the movie techniques of studying surfaces in 4-dimensions. He shows how to move among the standard examples of Klein bottles, and he gives a “movie move” decomposition of turning the 2-sphere inside out. In the final Chapter, higher dimensional spaces are examined from the same elementary point of view.The book is a guide book to a wide variety of topics. It will be of value to undergraduates who want to learn geometric topology and to graduate students who want examples with which they can make computations and who need an elementary description of topological spaces. Finally, the book should be interesting to other scientists and mathematicians who want to learn some examples of topological spaces.

This book constitutes a review volume on the relatively new subject of Quantum Topology. Quantum Topology has its inception in the 1984/1985 discoveries of new invariants of knots and links (Jones, Homfly and Kauffman polynomials). These invariants were rapidly connected with quantum groups and methods in statistical mechanics. This was followed by Edward Witten's introduction of methods of quantum field theory into the subject and the formulation by Witten and Michael Atiyah of the concept of topological quantum field theories.This book is a review volume of on-going research activity. The papers derive from talks given at the Special Session on Knot and Topological Quantum Field Theory of the American Mathematical Society held at Dayton, Ohio in the fall of 1992. The book consists of a self-contained article by Kauffman, entitled Introduction to Quantum Topology and eighteen research articles by participants in the special session.This book should provide a useful source of ideas and results for anyone interested in the interface between topology and quantum field theory.

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.

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.

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.

This book is about dealing with 3-manifolds using computers. Its emphasis is on presenting algorithms which are used for solving (in practice) the homeomorphism problem for the smallest of these objects. The key concept is the 3-gem, a special kind of edge-colored graph, which encodes the manifold via a ball complex. Passages between 3-gems and more standard presentations like Heegaard diagrams and surgery descriptions are provided. A catalogue of all closed orientable 3-manifolds induced by 3-gems up to 30 vertices is included. In order to help the classification, various invariants are presented, including the new quantum invariants.

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.

This volume includes both rigorous asymptotic results on the inevitability of random knotting and linking, and Monte Carlo simulations of knot probability at small lengths. The statistical mechanics and topology of surfaces on the d-dimensional simple cubic lattice are investigated. The energy of knots is studied both analytically and numerically. Vassiliev invariants are investigated and used in random knot simulations. A mutation scheme which leaves the Jones polynomial unaltered is described. Applications include the investigation of RNA secondary structure using Vassiliev invariants, and the direct experimental measurement of DNA knot probability as a function of salt concentration in random cyclization experiments on linear DNA molecules. The papers in this volume reflect the diversity of interest across science and mathematics in this subject, from topology to statistical mechanics to theoretical chemistry to wet-lab molecular biology.

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.

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced.In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space.Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface.Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view.This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced.In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space.Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface.Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view.This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.

A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many is: “Is there a ‘best’ bend and, if so, what is it?”Most of the well-known bends happen to be symmetric — that is, the two constituent cords within the bend have the same geometric shape and size, and interrelationship with the other. Such ‘symmetric bends’ have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage of being easier to tie (with less chance of error), and of probably being stronger, since neither end is the weaker.This book presents a mathematical theory of symmetric bends, together with a simple explanation of how such bends may be invented. Also discussed are the additionally symmetric ‘triply symmetric’ bends. Full details, including beautiful colour pictures, are given of the ‘best 60’ known symmetric bends, many of which were created by these methods of invention.This work will appeal to many — mathematicians as well as non-mathematicians interested in beautiful and useful knots.

This book presents two new mathematical techniques: nonstandard logics and nonstandard metrics. The techniques are applied to current problems in physics, i.e. the hidden variable problem, the local and nonlocal problems, etc.

The authors aim to reinstate a spirit of philosophical enquiry in physics. They abandon the intuitive continuum concepts and build up constructively a combinatorial mathematics of process. This radical change alone makes it possible to calculate the coupling constants of the fundamental fields which — via high energy scattering — are the bridge from the combinatorial world into dynamics. The untenable distinction between what is ‘observed’, or measured, and what is not, upon which current quantum theory is based, is not needed. If we are to speak of mind, this has to be present — albeit in primitive form — at the most basic level, and not to be dragged in at one arbitrary point to avoid the difficulties about quantum observation. There is a growing literature on information-theoretic models for physics, but hitherto the two disciplines have gone in parallel. In this book they interact vitally.

This book presents two new mathematical techniques: nonstandard logics and nonstandard metrics. The techniques are applied to current problems in physics, i.e. the hidden variable problem, the local and nonlocal problems, etc.

This book brings together twenty essays on diverse topics in the history and science of knots. It is divided into five parts, which deal respectively with knots in prehistory and antiquity, non-European traditions, working knots, the developing science of knots, and decorative and other aspects of knots.Its authors include archaeologists who write on knots found in digs of ancient sites (one describes the knots used by the recently discovered Ice Man); practical knotters who have studied the history and uses of knots at sea, for fishing and for various life support activities; a historian of lace; a computer scientist writing on computer classification of doilies; and mathematicians who describe the history of knot theories from the eighteenth century to the present day.In view of the explosion of mathematical theories of knots in the past decade, with consequential new and important scientific applications, this book is timely in setting down a brief, fragmentary history of mankind's oldest and most useful technical and decorative device — the knot.

This book constitutes a review volume on the relatively new subject of Quantum Topology. Quantum Topology has its inception in the 1984/1985 discoveries of new invariants of knots and links (Jones, Homfly and Kauffman polynomials). These invariants were rapidly connected with quantum groups and methods in statistical mechanics. This was followed by Edward Witten's introduction of methods of quantum field theory into the subject and the formulation by Witten and Michael Atiyah of the concept of topological quantum field theories.This book is a review volume of on-going research activity. The papers derive from talks given at the Special Session on Knot and Topological Quantum Field Theory of the American Mathematical Society held at Dayton, Ohio in the fall of 1992. The book consists of a self-contained article by Kauffman, entitled Introduction to Quantum Topology and eighteen research articles by participants in the special session.This book should provide a useful source of ideas and results for anyone interested in the interface between topology and quantum field theory.

This book sets up a discrete universe with minimum and maximum dimensions. Singularity is rejected.Entropic Spacetime Theory divides the universe into a kinetic system and an entropic spacetime. The kinetic system is what our present physics is all about; it deals with radiation (vector bosons) and mass particles (fermions). Relativity and quantum mechanics deal almost entirely in the kinetic system.The entropic spacetime (EST) defines space; in this theory there is no vacuum — EST is space. Made up of energy and dipole charges, its values can be converted into length and time.The theory offers a new description of space, a new cosmology, names space as the original creator of all new matter and radiation.

This book is about “diamond”, a logic of paradox. In diamond, a statement can be true yet false; an “imaginary” state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. Diamond is a new way to solve the dilemmas of higher mathematics. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book consists of two sections: Elementary; which covers the classic paradoxes of mathematical logic and shows how they can be resolved in this new system; and Advanced, which relates diamond to Boolean logic, three-valued logic, Gödelian meta-mathematics and dilemma games.

It now appears that the old argument about Lorentz vs Galileo relativity is passing into history. The Lorentz symmetry may soon become obsolete itself just as the Galileo symmetry did about 1900. The tremendous successes of QED represent real progress in our quest to understand nature. The answer is not to go as most “outsiders” but to go forward — beyond to new ideas and equations that will match nature even better than QED does. This book shows us a new view of relativity and quantum equations. It has new equations that extend Lorentz Maxwell and Dirac.

This volume consists of ten lectures given at an international workshop/conference on knot theory held in July 1996 at Waseda University Conference Center. It was organised by the International Research Institute of Mathematical Society of Japan. The workshop was attended by nearly 170 mathematicians from Japan and 14 other countries, most of whom were specialists in knot theory. The lectures can serve as an introduction to the field for advanced undergraduates, graduates and also researchers working in areas such as theoretical physics.

This book is about “delta”, a paradox logic. In delta, a statement can be true yet false; an intermediate state, midway between being and non-being. Delta's imaginary value solves many paradoxes unsolvable in two-valued Boolean logic, including Russell's, Cantor's, Berry's and Zeno's.Delta has three parts: “inner delta logic”, covering “Kleenean logic”, which resolves self-reference; outer delta logic, covering Z mod 3, conjugate logics, cyclic distribution, and the voter's paradox; and “beyond delta logic”, covering four-valued logic and games.

This book is based on the authors' research on rendering images of higher dimensional fractals by a distance estimation technique. It is self-contained, giving a careful treatment of both the known techniques and the authors' new methods. The distance estimation technique was originally applied to Julia sets and the Mandelbrot set in the complex plane. It was justified, through the work of Douady and Hubbard, by deep results in complex analysis. In this book the authors generalise the distance estimation to quaternionic and other higher dimensional fractals, including fractals derived from iteration in the Cayley numbers (octonionic fractals). The generalization is justified by new geometric arguments that circumvent the need for complex analysis. This puts on a firm footing the authors' present work and the second author's earlier work with John Hart and Dan Sandin. The results of this book will be of great interest to mathematicians and computer scientists interested in fractals and computer graphics.

In this book, experts in different fields of mathematics, physics, chemistry and biology present unique forms of knots which satisfy certain preassigned criteria relevant to a given field. They discuss the shapes of knotted magnetic flux lines, the forms of knotted arrangements of bistable chemical systems, the trajectories of knotted solitons, and the shapes of knots which can be tied using the shortest piece of elastic rope with a constant diameter.

One of the most significant unsolved problems in mathematics is the complete classification of knots. The main purpose of this book is to introduce the reader to the use of computer programming to obtain the table of knots. The author presents this problem as clearly and methodically as possible, starting from the very basics. Mathematical ideas and concepts are extensively discussed, and no advanced background is required.

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.

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.