Braid Foliations in Low-Dimensional Topology
About this Title
Douglas J. LaFountain, Western Illinois University, Macomb, IL and William W. Menasco, University at Buffalo, Buffalo, NY
Publication: Graduate Studies in Mathematics
Publication Year: 2017; Volume 185
ISBNs: 978-1-4704-3660-5 (print); 978-1-4704-4268-2 (online)
MathSciNet review: MR3702003
MSC: Primary 57M25; Secondary 20F36, 57M27, 57R17, 57R30
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.
Graduate students and researchers interested in geometry and topology.
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