Knots and Surfaces

Author: N. D. Gilbert,T. Porter

Publisher: Oxford University Press, UK

ISBN: 0191591904

Category: Knot theory

Page: 280

View: 5513

Completely up-to-date, illustrated throughout, and written in an accessible style, Knots and Surfaces is an account of the mathematical theory of knots and its interaction with related fields. This is an area of intense research activity, and this text provides the advanced undergraduate with a superb introduction to this exciting field. Beginning with a simple diagrammatic approach, the book proceeds through recent advances to areas of current research. Topics including topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations combine to form a coherent and highly developed theory with which to explore and explain the accessible and intuitive problems of knots and surfaces. - ;The main theme of this book is the mathematical theory of knots and its interaction with the theory of surfaces and of group presentations. Beginning with a simple diagrammatic approach to the study of knots, reflecting the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent advances in our understanding to areas of current research. Topics included are straightforward introductions to topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. These topics combine into a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces. Both as an introduction to several areas of prime importance to the development of pure mathematics today, and as an account of pure mathematics in action in an unusual context, this book presents novel challenges to students and other interested readers. -

Beginning Topology

Author: Sue E. Goodman

Publisher: American Mathematical Soc.

ISBN: 0821847961

Category: Mathematics

Page: 236

View: 7698

Beginning Topology is designed to give undergraduate students a broad notion of the scope of topology in areas of point-set, geometric, combinatorial, differential, and algebraic topology, including an introduction to knot theory. A primary goal is to expose students to some recent research and to get them actively involved in learning. Exercises and open-ended projects are placed throughout the text, making it adaptable to seminar-style classes. The book starts with a chapter introducing the basic concepts of point-set topology, with examples chosen to captivate students' imaginations while illustrating the need for rigor. Most of the material in this and the next two chapters is essential for the remainder of the book. One can then choose from chapters on map coloring, vector fields on surfaces, the fundamental group, and knot theory. A solid foundation in calculus is necessary, with some differential equations and basic group theory helpful in a couple of chapters. Topics are chosen to appeal to a wide variety of students: primarily upper-level math majors, but also a few freshmen and sophomores as well as graduate students from physics, economics, and computer science. All students will benefit from seeing the interaction of topology with other fields of mathematics and science; some will be motivated to continue with a more in-depth, rigorous study of topology.

Lecture Notes on Knot Invariants

Author: Weiping Li

Publisher: World Scientific

ISBN: 9814675989

Category: Mathematics

Page: 200

View: 1070

The volume is focused on the basic calculation skills of various knot invariants defined from topology and geometry. It presents the detailed Hecke algebra and braid representation to illustrate the original Jones polynomial (rather than the algebraic formal definition many other books and research articles use) and provides self-contained proofs of the Tait conjecture (one of the big achievements from the Jones invariant). It also presents explicit computations to the Casson–Lin invariant via braid representations. With the approach of an explicit computational point of view on knot invariants, this user-friendly volume will benefit readers to easily understand low-dimensional topology from examples and computations, rather than only knowing terminologies and theorems. Contents:Basic Knots, Links and Their EquivalencesBraids and LinksKnot and Link InvariantsJones PolynomialsCasson Type Invariants Readership: Undergraduate and graduate students interested in learning topology and low dimensional topology. Key Features:Applies a computational approach to understand knot invariants with geometric meaningsProvides a complete proof of Tait's conjectures from an original Jones polynomial definitionGives recent new knot invariants from the approach of algebraic geometry (characteristic variety)Readers will get a hands-on approach to the topological concepts and various invariant, instead of just knowing more fancy wordsKeywords:Knot Classifications;Tait Conjectures;Reidemeister Moves;Characterization of Braid Representation;Unknotting Number;Bridge Number;Linking Number;Crossing Number;Wirtinger Presentation;Magnus Representation;Twisted Alexander Polynomial;Hecke Algebra;Ocneanu Trace;Jones Polynomial;Kauffman Bracket;Casson Type Invariant

Survey on Knot Theory

Author: Akio Kawauchi

Publisher: Springer Science & Business Media

ISBN: 9783764351243

Category: Mathematics

Page: 420

View: 3872

Knot theory is a rapidly developing field of research with many applications not only for mathematics. The present volume, written by a well-known specialist, gives a complete survey of knot theory from its very beginnings to today's most recent research results. The topics include Alexander polynomials, Jones type polynomials, and Vassiliev invariants. With its appendix containing many useful tables and an extended list of references with over 3,500 entries it is an indispensable book for everyone concerned with knot theory. The book can serve as an introduction to the field for advanced undergraduate and graduate students. Also researchers working in outside areas such as theoretical physics or molecular biology will benefit from this thorough study which is complemented by many exercises and examples.

Knots and Links

Author: Peter R. Cromwell

Publisher: Cambridge University Press

ISBN: 9780521548311

Category: Mathematics

Page: 328

View: 4266

Knots and links are studied by mathematicians, and are also finding increasing application in chemistry and biology. Many naturally occurring questions are often simple to state, yet finding the answers may require ideas from the forefront of research. This readable and richly illustrated 2004 book explores selected topics in depth in a way that makes contemporary mathematics accessible to an undergraduate audience. It can be used for upper-division courses, and assumes only knowledge of basic algebra and elementary topology. Together with standard topics, the book explains: polygonal and smooth presentations; the surgery equivalence of surfaces; the behaviour of invariants under factorisation and the satellite construction; the arithmetic of Conway's rational tangles; arc presentations. Alongside the systematic development of the main theory, there are discussion sections that cover historical aspects, motivation, possible extensions, and applications. Many examples and exercises are included to show both the power and limitations of the techniques developed.

Introduction to 3-Manifolds

Author: Jennifer Schultens

Publisher: American Mathematical Soc.

ISBN: 1470410206

Category: Mathematics

Page: 286

View: 3150

This book grew out of a graduate course on 3-manifolds and is intended for a mathematically experienced audience that is new to low-dimensional topology. The exposition begins with the definition of a manifold, explores possible additional structures on manifolds, discusses the classification of surfaces, introduces key foundational results for 3-manifolds, and provides an overview of knot theory. It then continues with more specialized topics by briefly considering triangulations of 3-manifolds, normal surface theory, and Heegaard splittings. The book finishes with a discussion of topics relevant to viewing 3-manifolds via the curve complex. With about 250 figures and more than 200 exercises, this book can serve as an excellent overview and starting point for the study of 3-manifolds.

The classification of knots and 3-dimensional spaces

Author: Geoffrey Hemion

Publisher: Oxford University Press, USA


Category: Art

Page: 162

View: 7275

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.

Knots and quantum gravity

Author: John C. Baez

Publisher: Oxford University Press, USA


Category: Biography & Autobiography

Page: 229

View: 7663

This book is the first title in a new series from Oxford University Press; a series designed to make the most recent applied mathematics research more easily accessible to all professionals, postgraduates, and researchers pursuing a specific line of investigation. Recent work by mathematicians and physicists has uncovered surprising connections between knot theory and the problem of developing a quantum theory of gravity. This volume is the proceedings of a workshop held at the University of California at Riverside attended by many experts in this excitingarea of research. The purpose of the workshop was to bring together researchers in knot theory and quantum gravity and form more bridges between the two subjects. Most of the talks were given by researchers whose work has significance for both subjects. This volume contains expository papers as well as newresults, and should serve as a guide for mathematicians and physicists seeking to understand this rapidly developing area of research.

Curve and Surface Fitting with Splines

Author: Paul Dierckx

Publisher: Oxford University Press

ISBN: 9780198534402

Category: Computers

Page: 285

View: 7639

The fitting of a curve or surface through a set of observational data is a very frequent problem in different disciplines (mathematics, engineering, medicine, ...) with many interesting applications. This book describes the algorithms and mathematical fundamentals of a widely used softwarepackage for data fitting with (tensor product) splines. As such it gives a survey of possibilities and benefits but also of the problems to cope with when approximating with this popular type of function. In particular it is demonstrated in detail how the properties of B-splines can be fullyexploited for improving the computational efficiency and for incorporating different boundary or shape preserving constraints. Special attention is also paid to strategies for an automatic and adaptive knot selection with intent to obtain serious data reductions. The practical use of the smoothingsoftware is illustrated with many examples, academic as well as taken from real life.


Conference Proceedings, August 4-9, 1996

Author: Holly Rushmeier

Publisher: Addison-Wesley Professional

ISBN: 9780201948004

Category: Computers

Page: 528

View: 2658

These SIGGRAPH conference proceedings feature topical and current papers on computer graphics, desktop video and multimedia workstations. The CD-ROM contains the presentations from the conference workshops and lectures.

Discrete Integrable Geometry and Physics

Author: Alexander I. Bobenko,Ruedi Seiler

Publisher: Oxford University Press, USA

ISBN: 9780198501602

Category: Mathematics

Page: 370

View: 5886

Recent interactions between the fields of geometry, classical and quantum dynamical systems, and visualization of geometric objects such as curves and surfaces have led to the observation that most concepts of surface theory and of the theory of integrable systems have natural discreteanalogues. These are characterized by the property that the corresponding difference equations are integrable, and has led in turn to some important applications in areas of condensed matter physics and quantum field theory, amongst others. The book combines the efforts of a distinguished team ofauthors from various fields in mathematics and physics in an effort to provide an overview of the subject. The mathematical concepts of discrete geometry and discrete integrable systems are firstly presented as fundamental and valuable theories in themselves. In the following part these concepts areput into the context of classical and quantum dynamics.

If China Attacks Taiwan

Military Strategy, Politics and Economics

Author: Steve Tsang

Publisher: Routledge

ISBN: 1136916334

Category: Political Science

Page: 224

View: 774

This is a new analysis of the key issues facing Chinese policy makers in their approach towards Taiwan. This is one of the most tense and potentially explosive relationships in world politics. This book explains succinctly the impetus, the methods and the consequences if China is to use force, a prospect that has become greater following the return of President Chen Shui-bian to power in Taiwan for a second term in 2004. If China Attacks Taiwan shows how in reality there can be no real winner in such an eventuality and how the consequences would be dire not just for Taiwan and China, but East Asia as a whole. Whether China will use force depends ultimately on how its policy making apparatus assess potential US intervention, whether its armed forces can subdue Taiwan and counter US military involvement, as well as on its assessment of the likely consequences. Given the extremely high probability of American involvement this volume appeals to not only scholars and students working on China, its foreign policy and the security and prosperity of East Asia, but also to policy makers and journalists interested in China’s rise and its defense policy, Taiwan’s security and development, regional stability as well as US policy toward China and the East Asia region generally. This book is essential for understanding China’s efforts to achieve a ‘peaceful rise’, which requires it to transform itself into a global power not by the actual use of force but by diplomacy backed up by rapidly expanding military power. This book is an excellent resource for all students and scholars of military and security studies, Asian (China/Taiwan) studies and international relations