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Knot Theory (Mathematical Association of America Textbooks)
By Charles Livingston
- Publisher: The Mathematical Association of America
- Number Of Pages: 258
- Publication Date: 1996-09-05
- ISBN-10 / ASIN: 0883850273
- ISBN-13 / EAN: 9780883850275
- Binding: Hardcover
Product Description:
Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject - the Conway, Jones and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology.
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Functorial Knot Theory : Categories of Tangles, Coherence, Categorical Deformations and Topological Invariants
By David N. Yetter
- Publisher: World Scientific Publishing Company
- Number Of Pages: 236
- Publication Date: 2001-04
- ISBN-10 / ASIN: 9810244436
- ISBN-13 / EAN: 9789810244439
- Binding: Hardcover
Product Description:
Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structure naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory. This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations.
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Formal Knot Theory (Mathematical Notes, No. 30)
By Louis H. Kauffman
- Publisher: Princeton University Press
- Number Of Pages: 167
- Publication Date: 1983-10
- ISBN-10 / ASIN: 0691083363
- ISBN-13 / EAN: 9780691083360
- Binding: Paperback
Product Description:
This exploration of combinatorics and knot theory is geared toward advanced undergraduates and graduate students. The author draws upon his work as a topologist to illustrate the relationships between knot theory and statistical mechanics, quantum theory, and algebra, as well as the role of knot theory in combinatorics. 1983 edition. Includes 51 illustrations.
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Knot Theory and Its Applications (Modern Birkhäuser Classics)
By Kunio Murasugi
- Publisher: Birkhäuser Boston
- Number Of Pages: 342
- Publication Date: 2007-10-03
- ISBN-10 / ASIN: 081764718X
- ISBN-13 / EAN: 9780817647186
- Binding: Paperback
Product Description:
Knot theory is a concept in algebraic topology that has found applications to a variety of mathematical problems as well as to problems in computer science, biological and medical research, and mathematical physics. This book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fields.
The book contains most of the fundamental classical facts about the theory, such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials; also included are key newer developments and special topics such as chord diagrams and covering spaces. The work introduces the fascinating study of knots and provides insight into applications to such studies as DNA research and graph theory. In addition, each chapter includes a supplement that consists of interesting historical as well as mathematical comments.
The author clearly outlines what is known and what is not known about knots. He has been careful to avoid advanced mathematical terminology or intricate techniques in algebraic topology or group theory. There are numerous diagrams and exercises relating the material. The study of Jones polynomials and the Vassiliev invariants are closely examined.
"The book ...develops knot theory from an intuitive geometric-combinatorial point of view, avoiding completely more advanced concepts and techniques from algebraic topology...Thus the emphasis is on a lucid and intuitive exposition accessible to a broader audience... The book, written in a stimulating and original style, will serve as a first approach to this interesting field for readers with various backgrounds in mathematics, physics, etc. It is the first text developing recent topics as the Jones polynomial and Vassiliev invariants on a level accessible also for non-specialists in the field." -Zentralblatt Math
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Complex Topological K-Theory
Cambridge University Press | ISBN: 0521856345 | 2008-03-24 | PDF | 218 pages | 1.6 MB
Topological K-theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. No background in algebraic topology is assumed; the reader need only have taken the standard first courses in real analysis, abstract algebra, and point-set topology. The book begins with a detailed discussion of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of the most important theorems in the subject, such as the Bott periodicity theorem and the Thom isomorphism theorem. The multiplicative structure of K-theory and the Adams operations are also discussed and the final chapter details the construction and computation of characteristic classes. With every important aspect of the topic covered, and exercises at the end of each chapter, this is the definitive book for a first course in topological K-theory.
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