Knot Theory

Author: Charles Livingston

Publisher: Cambridge University Press

ISBN: 9780883850275

Category: Mathematics

Page: 240

View: 4500

This book uses only linear algebra and basic group theory to study the properties of knots.

A Survey of Knot Theory

Author: Akio Kawauchi

Publisher: Birkhäuser

ISBN: 3034892276

Category: Mathematics

Page: 423

View: 612

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 this theory from its very beginnings to today's most recent research results. An indispensable book for everyone concerned with knot theory.

The Knot Book

An Elementary Introduction to the Mathematical Theory of Knots

Author: Colin Conrad Adams

Publisher: American Mathematical Soc.

ISBN: 0821836781

Category: Mathematics

Page: 306

View: 5541

Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. The Knot Book is an introduction to this rich theory, starting from our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. The Knot Book is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in The Knot Book.

Mathematical Interest Theory

Author: Leslie Vaaler,James Daniel

Publisher: MAA

ISBN: 9780883857540

Category: Business & Economics

Page: 475

View: 9504

Mathematical Interest Theory gives an introduction to how investments grow over time in a mathematically precise manner. The emphasis is on practical applications that give the reader a concrete understanding of why the various relationships should be true. Among the modern financial topics introduced are: arbitrage, options, futures, and swaps. The content of the book, along with an understanding of probability, will provide a solid foundation for readers embarking on actuarial careers.Mathematical Interest Theory includes more than 240 carefully worked examples. There are over 430 problems, and numerical answers are included in an appendix. A companion student solution manual has detailed solutions to the odd-numbered problems. Key Features • Detailed instruction on how to use the Texas Instruments BA II Plus and BA II Plus professional calculators. • Examples are worked out with the problem and solution delineated so that the reader can think about the problem before reading the solution presented in the text • Key formulas, facts and algorithms placed in boxes so that they stand out in the text, and new terms printed in boldface as they are introduced • Descriptive titles are given for the examples in the book,( i.e., “Finding a(t) from ?t” or “Finding a bond's yield rate” )to help students skimming the book quickly find relevant material.• Exercises feature applied financial questions, • Writing activities for each chapter introduce each homework set.

Quandles

Author: Mohamed Elhamdadi, Sam Nelson

Publisher: American Mathematical Soc.

ISBN: 1470422131

Category: Knot theory

Page: 245

View: 4840

From prehistory to the present, knots have been used for purposes both artistic and practical. The modern science of Knot Theory has ramifications for biochemistry and mathematical physics and is a rich source of research projects for undergraduate and graduate students and professionals alike. Quandles are essentially knots translated into algebra. This book provides an accessible introduction to quandle theory for readers with a background in linear algebra. Important concepts from topology and abstract algebra motivated by quandle theory are introduced along the way. With elementary self-contained treatments of topics such as group theory, cohomology, knotted surfaces and more, this book is perfect for a transition course, an upper-division mathematics elective, preparation for research in knot theory, and any reader interested in knots.

Lie Groups

A Problem Oriented Introduction Via Matrix Groups

Author: Harriet Pollatsek

Publisher: MAA

ISBN: 9780883857595

Category: Mathematics

Page: 177

View: 7434

The work of the Norwegian mathematician Sophus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires extensive mathematical prerequisites beyond the reach of the typical undergraduate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and important material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world.Lie Groups is an active learning text that can be used by students with a range of backgrounds and interests. The material is developed through 200 carefully chosen problems. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers.

Knots and Surfaces

A Guide to Discovering Mathematics

Author: David W. Farmer,Theodore B. Stanford

Publisher: American Mathematical Soc.

ISBN: 0821804510

Category: Mathematics

Page: 101

View: 9826

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 non science students who want to fulfill a science requirement. Also available from the AMS by David W. Farmer is ""Groups and Symmetry: A Guide to Discovering Mathematics"".

A TeXas Style Introduction to Proof

Author: Ron Taylor,Patrick X. Rault

Publisher: The Mathematical Association of America

ISBN: 1939512131

Category: Mathematics

Page: 176

View: 434

A TeXas Style Introduction to Proof is an IBL textbook designed for a one-semester course on proofs (the "bridge course") that also introduces TeX as a tool students can use to communicate their work. As befitting "textless" text, the book is, as one reviewer characterized it, "minimal." Written in an easy-going style, the exposition is just enough to support the activities, and it is clear, concise, and effective. The book is well organized and contains ample carefully selected exercises that are varied, interesting, and probing, without being discouragingly difficult.

Applications of Knot Theory

American Mathematical Society, Short Course, January 4-5, 2008, San Diego, California

Author: American Mathematical Society. Short course,Dorothy Buck

Publisher: American Mathematical Soc.

ISBN: 0821844660

Category: Mathematics

Page: 186

View: 8636

Over the past 20-30 years, knot theory has rekindled its historic ties with biology, chemistry, and physics as a means of creating more sophisticated descriptions of the entanglements and properties of natural phenomena--from strings to organic compounds to DNA. This volume is based on the 2008 AMS Short Course, Applications of Knot Theory. The aim of the Short Course and this volume, while not covering all aspects of applied knot theory, is to provide the reader with a mathematical appetizer, in order to stimulate the mathematical appetite for further study of this exciting field. No prior knowledge of topology, biology, chemistry, or physics is assumed. In particular, the first three chapters of this volume introduce the reader to knot theory (by Colin Adams), topological chirality and molecular symmetry (by Erica Flapan), and DNA topology (by Dorothy Buck). The second half of this volume is focused on three particular applications of knot theory. Louis Kauffman discusses applications of knot theory to physics, Nadrian Seeman discusses how topology is used in DNA nanotechnology, and Jonathan Simon discusses the statistical and energetic properties of knots and their relation to molecular biology.

Topology Now!

Author: Robert Messer,Philip Straffin

Publisher: MAA

ISBN: 9780883857441

Category: Mathematics

Page: 240

View: 7432

An undergraduate textbook on topology designed to have very few prerequisites.

Knot Theory and Its Applications

Author: Kunio Murasugi

Publisher: Springer Science & Business Media

ISBN: 0817647198

Category: Mathematics

Page: 341

View: 1979

This book introduces the study of knots, providing insights into recent applications in DNA research and graph theory. It sets forth fundamental facts such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials. It also covers more recent developments and special topics, such as chord diagrams and covering spaces. The author avoids advanced mathematical terminology and intricate techniques in algebraic topology and group theory. Numerous diagrams and exercises help readers understand and apply the theory. Each chapter includes a supplement with interesting historical and mathematical comments.

Category Theory and Applications

A Textbook for Beginners

Author: Marco Grandis

Publisher: World Scientific

ISBN: 9813231084

Category:

Page: 304

View: 9577

Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots. This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: Introduction Categories, Functors and Natural Transformations Limits and Colimits Adjunctions and Monads Applications in Algebra Applications in Topology and Algebraic Topology Applications in Homological Algebra Hints at Higher Dimensional Category Theory References Indices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topology The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints Three chapters and some extra sections are devoted to applications

Knots and Links

Author: Dale Rolfsen

Publisher: American Mathematical Soc.

ISBN: 0821834363

Category: Mathematics

Page: 439

View: 4875

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. Now back in print by the AMS, the book is still a standard reference in knot theory. It is written in a remarkable style that makes it useful for both beginners and researchers. Particularly noteworthy is the table of knots and links at the end. This volume is an excellent introduction to the topic and is suitable as a textbook for a course in knot theory or 3-manifolds. Other key books of interest on this topic available from the AMS are ""The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes"" and ""The Knot Book"".

Teaching and Learning of Knot Theory in School Mathematics

Author: Akio Kawauchi,Tomoko Yanagimoto

Publisher: Springer Science & Business Media

ISBN: 4431541381

Category: Mathematics

Page: 188

View: 3268

​This book is the result of a joint venture between Professor Akio Kawauchi, Osaka City University, well-known for his research in knot theory, and the Osaka study group of mathematics education, founded by Professor Hirokazu Okamori and now chaired by his successor Professor Tomoko Yanagimoto, Osaka Kyoiku University. The seven chapters address the teaching and learning of knot theory from several perspectives. Readers will find an extremely clear and concise introduction to the fundamentals of knot theory, an overview of curricular developments in Japan, and in particular a series of teaching experiments at all levels which not only demonstrate the creativity and the professional expertise of the members of the study group, but also give a lively impression of students’ learning processes. In addition the reports show that elementary knot theory is not just a preparation for advanced knot theory but also an excellent means to develop spatial thinking. The book can be highly recommended for several reasons: First of all, and that is the main intention of the book, it serves as a comprehensive text for teaching and learning knot theory. Moreover it provides a model for cooperation between mathematicians and mathematics educators based on substantial mathematics. And finally it is a thorough introduction to the Japanese art of lesson studies–again in the context of substantial mathematics.

An Invitation to Knot Theory

Virtual and Classical

Author: Heather A. Dye

Publisher: CRC Press

ISBN: 1498701655

Category: Mathematics

Page: 256

View: 9772

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra. The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

The Geometry and Physics of Knots

Author: Michael Francis Atiyah

Publisher: Cambridge University Press

ISBN: 9780521395540

Category: Mathematics

Page: 78

View: 8493

Deals with an area of 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 reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.

Grid Homology for Knots and Links

Author: Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó

Publisher: American Mathematical Soc.

ISBN: 1470417375

Category: Homology theory

Page: 410

View: 9604

Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

The Interface of Knots and Physics

American Mathematical Society Short Course, January 2-3, 1995, San Francisco, California

Author: Louis H. Kauffman,American Mathematical Society

Publisher: American Mathematical Soc.

ISBN: 0821803808

Category: Science

Page: 208

View: 8214

This book is the result of an AMS Short Course on Knots and Physics that was held in San Francisco (January 1994). The range of the course went beyond knots to the study of invariants of low dimensional manifolds and extensions of this work to four manifolds and to higher dimensions. The authors use ideas and methods of mathematical physics to extract topological information about knots and manifolds. Features: A basic introduction to knot polynomials in relation to statistical link invariants. Concise introductions to topological quantum field theories and to the role of knot theory in quantum gravity. Knots and Physics would be an excellent supplement to a course on algebraic topology or a physics course on field theory.

Subfactors and Knots

Author: Vaughan F. R. Jones,Conference Board of the Mathematical Sciences

Publisher: American Mathematical Soc.

ISBN: 0821807293

Category: Mathematics

Page: 113

View: 7979

This book is based on a set of lectures presented by the author at the NSF-CBMS Regional Conference, Applications of Operator Algebras to Knot Theory and Mathematical Physics, held at the U.S. Naval Academy in Annapolis in June 1988. The audience consisted of low-dimensional topologists and operator algebraists, so the speaker attempted to make the material comprehensible to both groups. He provides an extensive introduction to the theory of von Neumann algebras and to knot theory and braid groups. The presentation follows the historical development of the theory of subfactors and the ensuing applications to knot theory, including full proofs of some of the major results. The author treats in detail the Homfly and Kauffman polynomials, introduces statistical mechanical methods on knot diagrams, and attempts an analogy with conformal field theory. Written by one of the foremost mathematicians of the day, this book will give readers an appreciation of the unexpected interconnections between different parts of mathematics and physics.

Knots, Molecules, and the Universe

Author: Erica Flapan

Publisher: American Mathematical Soc.

ISBN: 1470425351

Category: Algebraic topology

Page: 386

View: 1456

This book is an elementary introduction to geometric topology and its applications to chemistry, molecular biology, and cosmology. It does not assume any mathematical or scientific background, sophistication, or even motivation to study mathematics. It is meant to be fun and engaging while drawing students in to learn about fundamental topological and geometric ideas. Though the book can be read and enjoyed by nonmathematicians, college students, or even eager high school students, it is intended to be used as an undergraduate textbook. The book is divided into three parts corresponding to the three areas referred to in the title. Part 1 develops techniques that enable two- and three-dimensional creatures to visualize possible shapes for their universe and to use topological and geometric properties to distinguish one such space from another. Part 2 is an introduction to knot theory with an emphasis on invariants. Part 3 presents applications of topology and geometry to molecular symmetries, DNA, and proteins. Each chapter ends with exercises that allow for better understanding of the material. The style of the book is informal and lively. Though all of the definitions and theorems are explicitly stated, they are given in an intuitive rather than a rigorous form, with several hundreds of figures illustrating the exposition. This allows students to develop intuition about topology and geometry without getting bogged down in technical details.