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

• Braids
• Braid automorphisms
• Classical links
• Braid presentation of links
• Deformation chain and Markov's theorem

• Surface links
• Surface link diagrams
• Motion pictures
• Normal forms of surface links
• Examples (Spinning)
• Ribbon surface links
• Presentations of surface link groups

• Branched coverings
• Surface braids
• Products of surface braids
• Braided surfaces
• Braid monodromy
• Chart descriptions
• Non-simple surface braids
• 1-handle surgery on surface braids

• 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

• Knot groups
• Unknotted surface braids and surface links
• Ribbon surface braids and surface links
• 3-braid 2-knots
• Unknotting surface braids and surface links
• Seifert algorithm for surface braids
• Basic symmetries in chart descriptions
• Singular surface braids and surface links
• Bibliography
• Index