Introduction Braid theory is a beautiful subject which combines the visual appeal and in- sights of topology with the precision and power of algebra. It is relevant not only to algebraists and topologists, but also to scientists working in many disciplines. It even touches upon such diverse fields as polymer chemistry, molecular biology, cryptography and robotics. The theory of braids has been an exceptionally active mathematical subject in recent decades. The field really caught fire in the mid 1980’s with the revo- lutionary discoveries of Vaughan Jones [115], providing strong connections with operator theory, statistical mechanics and utilizing many ideas which originated from mathematical physics. That braids have a natural ordering, compatible with their algebraic structure, was discovered a decade later by one of the authors (P.D.), and since then it has been intensively studied and generalized by many mathematicians, including the authors. That phenomenon is the subject of this book. One of the exciting aspects of this work is the rich variety of mathematical techniques that come into play. In these pages, one will find subtle combinatorics, applications of hyperbolic geometry, automata theory, laminations and triangu- lations, dynamics, even unprovability results, in addition to the more traditional methods of topology and algebra. A meeting of two classical subjects It was an idea whose time was overdue—the marriage of braid theory with the theory of orderable groups. The braid groups Bn were introduced by Emil Artin [4] in 1925—see also [5]. Indeed, many of the ideas date back to the nineteenth century in the works of Hurwicz, Klein, Poincar´ e, Riemann, and certainly other authors. One can even find a braid sketched in the notebooks of Gauss [97]—see [177] for a discussion about Gauss and braids, including a reproduction of the picture he drew in his notebook. The n-strand braid group Bn has a well-known presentation—other definitions will be given later: Bn = σ1, ..., σn−1 | σiσj = σjσi for |i j| 2, σiσjσi = σjσiσj for |i j| = 1 . We use B+ n for the monoid with the above presentation, which is called the n-strand braid monoid. The monoid Bn + is included in a larger submonoid Bn∗ + of Bn, called the dual braid monoid, which is associated with the presentation of Bn given by Birman, Ko and Lee in [15]—details may be found in Chapter VIII. To each braid, there is an associated permutation of the set {1, ..., n}, with σi sent to (i, i + 1), defining a homomorphism of Bn onto the symmetric group Sn. The kernel of this mapping is the pure braid group PBn. 1
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