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.
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 Non-simple surface braids 1-handle 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 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.