Braid Foliations in Low-Dimensional Topology: (Graduate Studies in Mathematics)

This book is a self-contained introduction to braid foliation techniques, which is a theory developed to study knots, links and surfaces in general 3-manifolds and more specifically in contact 3-manifolds. With style and content accessible to beginning students interested in geometric topology, each chapter centers around a key theorem or theorems. The particular braid foliation techniques needed to prove these theorems are introduced in parallel, so that the reader has an immediate "take-home" for the techniques involved. The reader will learn that braid foliations provide a flexible toolbox capable of proving classical results such as Markov's theorem for closed braids and the transverse Markov theorem for transverse links, as well as recent results such as the generalized Jones conjecture for closed braids and the Legendrian grid number conjecture for Legendrian links. Connections are also made between the Dehornoy ordering of the braid groups and braid foliations on surfaces. All of this is accomplished with techniques for which only mild prerequisites are required, such as an introductory knowledge of knot theory and differential geometry. The visual flavor of the arguments contained in the book is supported by over 200 figures.

Table of Contents:
Links and closed braids Braid foliations and Markov's theorem Exchange moves and Jones' conjecture Transverse links and Bennequin's inequality The transverse Markov theorem and simplicity Botany of braids and transverse knots Flypes and transverse nonsimplicity Arc presentations of links and braid foliations Braid foliations and Legendrian links Braid foliations and braid groups Open book foliations Braid foliations and convex surface theory Bibliography Index.

About the Author :
Douglas J. LaFountain, Western Illinois University, Macomb, IL. William W. Menasco, University at Buffalo, NY.

Review :
This research monograph is a highly readable and pleasant introduction to the toolkits that the authors call braid foliation techniques, a small but relatively underdeveloped corner of low-dimensional topology and geometry. It is written at a level that will be accessible to graduate students and researchers and is carefully structured and filled with useful examples." — J.S. Birman, Mathematical Reviews "The AMS once more presents the mathematical community with a strong text geared to getting graduate students and other relative beginners into the game. The present book is thorough and well-structured, leads the reader pretty deeply into the indicated parts of knot and link-theory and low-dimensional topology and does so effectively and (as far as I can tell) rather painlessly...All in all, the book looks like a hit." — Michael Berg, MAA Reviews