Preface The present volume follows a book, “Why are braids orderable?”, written by the same authors and published in 2002 by the Soci´ et´ e Math´ ematique de France in the series Panoramas et Synth` eses. We emphasize that this is not a new edition of that book. Although this book contains most of the material in the previous book, it also contains a considerable amount of new material. In addition, much of the original text has been completely rewritten, with a view to making it more readable and up-to-date. We have been able not only to include ideas that were unknown in 2002, but we have also benefitted from helpful comments by colleagues and students regarding the contents of the SMF book, and we have taken their advice to heart in writing this book. The reader is assumed to have some basic background in group theory and topology. However, we have attempted to make the ideas in this volume accessible and interesting to students and seasoned professionals alike. In fact, the question “Why are braids orderable?” has not been answered to our satisfaction, either in the book with that title or the present volume. That is, we do not understand precisely what makes the braid groups so special that they enjoy an ordering so easy to describe, so challenging to construct and with such subtle properties as are described in these pages. The best we can offer is some insight into the easier question, “How are braids orderable?” Patrick Dehornoy, Caen Ivan Dynnikov, Moscow Dale Rolfsen, Vancouver Bert Wiest, Rennes December 2007 ix
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