**Graduate Studies in Mathematics**

Volume: 185;
2017;
304 pp;
Hardcover

MSC: Primary 57; 20;

Print ISBN: 978-1-4704-3660-5

Product Code: GSM/185

List Price: $83.00

Individual Member Price: $66.40

**Electronic ISBN: 978-1-4704-4268-2
Product Code: GSM/185.E**

List Price: $83.00

Individual Member Price: $66.40

#### Supplemental Materials

# Braid Foliations in Low-Dimensional Topology

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*Douglas J. LaFountain; William W. Menasco*

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.

#### Readership

Graduate students and researchers interested in geometry and topology.

#### Table of Contents

# Table of Contents

## Braid Foliations in Low-Dimensional Topology

- Cover Cover11
- Title page i2
- Contents v6
- Preface ix10
- Chapter 1. Links and closed braids 114
- Chapter 2. Braid foliations and Markov’s theorem 2134
- Chapter 3. Exchange moves and Jones’ conjecture 5366
- 3.1. Valence-two elliptic points 5467
- 3.2. Identifying exchange moves 5669
- 3.3. Reducing valence of elliptic points with changes of foliation 6376
- 3.4. Jones’ conjecture and the generalized Jones conjecture 6780
- 3.5. Stabilizing to embedded annuli 6881
- 3.6. Euler characteristic calculations 7285
- 3.7. Proof of the generalized Jones conjecture 7689
- Exercises 8093

- Chapter 4. Transverse links and Bennequin’s inequality 8598
- 4.1. Calculating the writhe and braid index 8598
- 4.2. The standard contact structure and transverse links 88101
- 4.3. The characteristic foliation and Giroux’s elimination lemma 92105
- 4.4. Transverse Alexander theorem 95108
- 4.5. The self-linking number and Bennequin’s inequality 97110
- 4.6. Tight versus overtwisted contact structures 101114
- 4.7. Transverse link invariants in low-dimensional topology 103116
- Exercises 104117

- Chapter 5. The transverse Markov theorem and simplicity 107120
- Chapter 6. Botany of braids and transverse knots 137150
- Chapter 7. Flypes and transverse non-simplicity 151164
- Chapter 8. Arc presentations of links and braid foliations 167180
- Chapter 9. Braid foliations and Legendrian links 187200
- 9.1. Legendrian links in the standard contact structure 187200
- 9.2. The Thurston-Bennequin and rotation numbers 191204
- 9.3. Legendrian links and grid diagrams 194207
- 9.4. Mirrors, Legendrian links and the grid number conjecture 198211
- 9.5. Steps 1 and 2 in the proof of Theorem 9.8 201214
- 9.6. Braided grid diagrams, braid foliations and destabilizations 204217
- 9.7. Step 3 in the proof of Theorem 9.8 210223
- Exercises 217230

- Chapter 10. Braid foliations and braid groups 219232
- 10.1. The braid group 𝐵_{𝑛} 219232
- 10.2. The Dehornoy ordering on the braid group 221234
- 10.3. Braid moves and the Dehornoy ordering 223236
- 10.4. The Dehornoy floor and braid foliations 225238
- 10.5. Band generators and the Dehornoy ordering 231244
- 10.6. Dehornoy ordering, braid foliations and knot genus 233246
- Exercises 236249

- Chapter 11. Open book foliations 239252
- 11.1. Open book decompositions of 3-manifolds 239252
- 11.2. Open book foliations 241254
- 11.3. Markov’s theorem in open books 242255
- 11.4. Change of foliation and exchange moves in open books 245258
- 11.5. Contact structures and open books 248261
- 11.6. The fractional Dehn twist coefficient 249262
- 11.7. Planar open book foliations and a condition on FDTC 253266
- 11.8. A generalized Jones conjecture for certain open books 257270
- Exercises 260273

- Chapter 12. Braid foliations and convex surface theory 263276
- Bibliography 281294
- Index 287300
- Back Cover Back Cover1305