Cover

Title page

Contents

Preface

Chapter 1. Links and closed braids

1.1. Links

1.2. Closed braids and Alexander’s theorem

1.3. Braid index and writhe

1.4. Stabilization, destabilization and exchange moves

1.5. Braid groups

1.6. Varying perspectives of closed braids

Exercises

Chapter 2. Braid foliations and Markov’s theorem

2.1. Two examples

2.2. Braid foliation basics

2.3. Obtaining braid foliations with only arcs

2.4. Identifying destabilizations and stabilizations

2.5. Markov’s theorem for the unlink

2.6. Annuli cobounded by two braids

2.7. Markov’s theorem

Exercises

Chapter 3. Exchange moves and Jones’ conjecture

3.1. Valence-two elliptic points

3.2. Identifying exchange moves

3.3. Reducing valence of elliptic points with changes of foliation

3.4. Jones’ conjecture and the generalized Jones conjecture

3.5. Stabilizing to embedded annuli

3.6. Euler characteristic calculations

3.7. Proof of the generalized Jones conjecture

Exercises

Chapter 4. Transverse links and Bennequin’s inequality

4.1. Calculating the writhe and braid index

4.2. The standard contact structure and transverse links

4.3. The characteristic foliation and Giroux’s elimination lemma

4.4. Transverse Alexander theorem

4.5. The self-linking number and Bennequin’s inequality

4.6. Tight versus overtwisted contact structures

4.7. Transverse link invariants in low-dimensional topology

Exercises

Chapter 5. The transverse Markov theorem and simplicity

5.1. Transverse isotopies

5.2. Transverse Markov theorem

5.3. Exchange reducibility implies transverse simplicity

5.4. The unlink is transversely simple

5.5. Torus knots are transversely simple

Exercises

Chapter 6. Botany of braids and transverse knots

6.1. Infinitely many conjugacy classes

6.2. Finitely many exchange equivalence classes

6.3. Finitely many transverse isotopy classes

6.4. Exotic botany and open questions

Exercises

Chapter 7. Flypes and transverse non-simplicity

7.1. Flype templates

7.2. Botany of 3-braids

7.3. The clasp annulus revisited

7.4. A weak MTWS for 3-braids

7.5. Transverse isotopies and a transverse clasp annulus

7.6. Transversely non-simple 3-braids

Exercises

Chapter 8. Arc presentations of links and braid foliations

8.1. Arc presentations and grid diagrams

8.2. Basic moves for arc presentations

8.3. Arc presentations and braid foliations

8.4. Arc presentations of the unknot and braid foliations

8.5. Monotonic simplification of the unknot

Exercises

Chapter 9. Braid foliations and Legendrian links

9.1. Legendrian links in the standard contact structure

9.2. The Thurston-Bennequin and rotation numbers

9.3. Legendrian links and grid diagrams

9.4. Mirrors, Legendrian links and the grid number conjecture

9.5. Steps 1 and 2 in the proof of Theorem 9.8

9.6. Braided grid diagrams, braid foliations and destabilizations

9.7. Step 3 in the proof of Theorem 9.8

Exercises

Chapter 10. Braid foliations and braid groups

10.1. The braid group 𝐵_{𝑛}

10.2. The Dehornoy ordering on the braid group

10.3. Braid moves and the Dehornoy ordering

10.4. The Dehornoy floor and braid foliations

10.5. Band generators and the Dehornoy ordering

10.6. Dehornoy ordering, braid foliations and knot genus

Exercises

Chapter 11. Open book foliations

11.1. Open book decompositions of 3-manifolds

11.2. Open book foliations

11.3. Markov’s theorem in open books

11.4. Change of foliation and exchange moves in open books

11.5. Contact structures and open books

11.6. The fractional Dehn twist coefficient

11.7. Planar open book foliations and a condition on FDTC

11.8. A generalized Jones conjecture for certain open books

Exercises

Chapter 12. Braid foliations and convex surface theory

12.1. Convex surfaces in contact 3-manifolds

12.2. Dividing sets for convex surfaces

12.3. Bypasses for convex surfaces

12.4. Non-thickenable solid tori

12.5. Exotic botany and Legendrian invariants

Exercises

Bibliography

Index

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