The Calculus of Braids

This introduction to braid groups keeps prerequisites to a minimum, while discussing their rich mathematical properties and applications.

Patrick Dehornoy (Author)

9781108925860, Cambridge University Press

Paperback / softback, published 9 September 2021

260 pages
23 x 15 x 1.6 cm, 0.4 kg

'… 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

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.

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.

Subject Areas: Algebraic topology [PBPD], Calculus & mathematical analysis [PBK], Algebra [PBF]