The Calculus of Braids

Patrick Dehornoy

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English

Cambridge University Press
09 September 2021

Algebra; Calculus & mathematical analysis; Algebraic topology

Series: London Mathematical Society Student Texts

Everyone knows what braids are, whether they be made of hair, knitting wool, or electrical cables. However, it is not so evident that we can construct a theory about them, i.e. to elaborate a coherent and mathematically interesting corpus of results concerning them. This book demonstrates that there is a resoundingly positive response to this question: braids are fascinating objects, with a variety of rich mathematical properties and potential applications. A special emphasis is placed on the algorithmic aspects and on what can be called the 'calculus of braids', in particular the problem of isotopy. Prerequisites are kept to a minimum, with most results being established from scratch. An appendix at the end of each chapter gives a detailed introduction to the more advanced notions required, including monoids and group presentations. Also included is a range of carefully selected exercises to help the reader test their knowledge, with solutions available.

By: Patrick Dehornoy
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 230mm, Width: 150mm, Spine: 16mm
Weight: 400g
ISBN: 9781108925860
ISBN 10: 1108925863
Series: London Mathematical Society Student Texts
Pages: 220
Publication Date: 09 September 2021
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

1. Geometric braids; 2. Braid groups; 3. Braid monoids; 4. The greedy normal form; 5. The Artin representation; 6. Handle reduction; 7. The Dynnikov coordinates; 8. A few avenues of investigation; 9. Solutions to the exercises; Glossary; References; Index.

Patrick Dehornoy (1952–2019) was Professor Emeritus at Université Caen Normandie, and Senior Member Emeritus of the Institut Universitaire de France. His research focused on set theory, algebra, and topology in small dimensions. He authored close to 100 research articles and eight books.

Reviews for The Calculus of Braids

'This book is a very clear introduction to the theory of braids, addressed to beginners. The author has made a great effort to render the exposition in this book pleasant and interesting and to make the proofs complete. This book is probably the best first introduction to the theory of braids.' Athanase Papadopoulos, zbMATH '… this text is an excellent reference for graduate students and researchers. It is great to have a single, comprehensive resource addressing the braid isotopy problem using so many different approaches. The definitions are thoughtfully constructed and proofs are very thorough and detailed.' Katherine Vance, MAA Reviews