Knots and Surfaces (Oxford Science Publications) by N. D. Gilbert (PDF)

Description

This highly readable text details the interaction between the mathematical theory of knots and the theories of surfaces and group presentations. It expertly introduces several topics critical to the development of pure mathematics while providing an account of math “in action” in an unusual context. Beginning with a simple diagrammatic approach to the study of knots that reflects the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent research advances. Topics include topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. The authors skillfully combine these topics to form a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces to students and researchers in mathematics.

User’s Reviews

Editorial Reviews: Review “This excellent book is based….on a third-year undergraduate course….The authors have been successful in giving as much motivation as possible for each of the notions considered in the book.” –Mathematical Reviews From the Back Cover The main theme of this book is the mathematical theory of knots and its interaction with the theory of surfaces. 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. 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, the book presents novel challenges to students and other interested readers. About the Author N. D. Gilbert is at Heriot-Watt University, Edinburgh. T. Porter is at University of Wales at Bangor. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐It’s a well-written book, balancing rigor and informality, perhaps with more of a concession to the latter. This is appropriate for an introductory text, however. It has excellent exercises and comments, and it’s just plain fun to interact with the text.

⭐This unique and vibrant book is an introductory book on knot theory that somehow sneaks in a lot more rigorous mathematics than you would expect. Without seeming overly difficult, it somehow manages to include a concise introduction to the basics of knot theory, basic topology, and even a little bit of graph theory, algebraic topology and the necessary algebra.While assuming little background, this book covers an extraordinarily diverse range of material, and ties it all together. The book is written so that a third-year undergraduate could understand it, but it’s interesting enough that a graduate student will still find it fascinating.What I love most about this book is the choice and ordering of topics–the authors dive right into the material, going to some depth in exploring polynomial invariants before they even touch any “abstract nonsense” so to speak; the machinery is developed throughout the book, as it is needed, and as a result seems natural and fully motivated.I think this book is excellent for self-study; it would also make a great textbook for a course, although to some extent the material in the course would be dictated by what the book covers. I also think that someone teaching a topology or graph theory courses should keep this book in mind and recommend it to any students inquiring about connections to knot theory.

⭐I loved the book. It is easy to read and it covers a lot of material.It is a great introductory book to knot theory: Alexander polynomial, Jones polynomial, Kauffman bracket, HOMFLY polynomial… It also gives some bases in groupe theory (group presentations) and algebraic topology (fundamental group, Van Kampen theorem).

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