Mathematical Surveys and Monographs 2002; 313 pp; hardcover Volume: 95 ISBN10: 0821829696 ISBN13: 9780821829691 List Price: US$91 Member Price: US$72.80 Order Code: SURV/95
See also: Ordering Braids  Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen and Bert Wiest  Braid theory and knot theory are related via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus, one can use braid theory to study knot theory and vice versa. In this book, the author generalizes braid theory to dimension four. He develops the theory of surface braids and applies it to study surface links. In particular, the generalized Alexander and Markov theorems in dimension four are given. This book is the first to contain a complete proof of the generalized Markov theorem. Surface links are studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. Included is a table of knotted surfaces with a computation of Alexander polynomials. Braid techniques are extended to represent link homotopy classes. The book is geared toward a wide audience, from graduate students to specialists. It would make a suitable text for a graduate course and a valuable resource for researchers. Readership Graduate students and research mathematicians interested in manifolds, cell complexes, and group theory and generalizations. Reviews "This book presents this surface braid theory in a systematic and well organized manner, and is the first to overview the theory ... A complete proof of an analogue of Markov's theorem is presented, which is made available in print for the first time in this book ... Throughout the book, the description of the material is concise and precise, and illustrations are effective and helpful."  Mathematical Reviews "The present book gives the only full treatment of the basic results on surface braids, and is likely to become the standard reference for its topic."  Zentralblatt MATH Table of Contents  Basic notions and notation
Classical braids and links  Braids
 Braid automorphisms
 Classical links
 Braid presentation of links
 Deformation chain and Markov's theorem
Surface knots and links  Surface links
 Surface link diagrams
 Motion pictures
 Normal forms of surface links
 Examples (Spinning)
 Ribbon surface links
 Presentations of surface link groups
Surface braids  Branched coverings
 Surface braids
 Products of surface braids
 Braided surfaces
 Braid monodromy
 Chart descriptions
 Nonsimple surface braids
 1handle surgery on surface braids
Braid presentation of surface links  The normal braid presentation
 Braiding ribbon surface links
 Alexander's theorem in dimension four
 Split union and connected sum
 Markov's theorem in dimension four
 Proof of Markov's theorem in dimension four
Surface braids and surface links  Knot groups
 Unknotted surface braids and surface links
 Ribbon surface braids and surface links
 3braid 2knots
 Unknotting surface braids and surface links
 Seifert algorithm for surface braids
 Basic symmetries in chart descriptions
 Singular surface braids and surface links
 Bibliography
 Index
