**Mathematical Surveys and Monographs**

Volume: 95;
2002;
313 pp;
Hardcover

MSC: Primary 57;
Secondary 20

Print ISBN: 978-0-8218-2969-1

Product Code: SURV/95

List Price: $96.00

Individual Member Price: $76.80

**Electronic ISBN: 978-1-4704-1322-4
Product Code: SURV/95.E**

List Price: $96.00

Individual Member Price: $76.80

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# Braid and Knot Theory in Dimension Four

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*Seiichi Kamada*

Braid theory and knot theory are related via two famous results due to
Alexander and Markov. Alexander's theorem states that any knot or link can be
put into braid form. Markov's theorem gives necessary and sufficient conditions
to conclude that two braids represent the same knot or link. Thus, one can use
braid theory to study knot theory and vice versa.

In this book, the author generalizes braid theory to dimension four. He
develops the theory of surface braids and applies it to study surface links. In
particular, the generalized Alexander and Markov theorems in dimension four are
given. This book is the first to contain a complete proof of the generalized
Markov theorem.

Surface links are studied via the motion picture method, and some important
techniques of this method are studied. For surface braids, various methods to
describe them are introduced and developed: the motion picture method, the
chart description, the braid monodromy, and the braid system. These tools are
fundamental to understanding and computing invariants of surface braids and
surface links.

Included is a table of knotted surfaces with a computation of Alexander
polynomials. Braid techniques are extended to represent link homotopy classes.
The book is geared toward a wide audience, from graduate students to
specialists. It would make a suitable text for a graduate course and a valuable
resource for researchers.

#### Readership

Graduate students and research mathematicians interested in manifolds, cell complexes, and group theory and generalizations.

#### Reviews & Endorsements

This book presents this surface braid theory in a systematic and well organized manner, and is the first to overview the theory … A complete proof of an analogue of Markov's theorem is presented, which is made available in print for the first time in this book … Throughout the book, the description of the material is concise and precise, and illustrations are effective and helpful.

-- Mathematical Reviews

The present book gives the only full treatment of the basic results on surface braids, and is likely to become the standard reference for its topic.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Braid and Knot Theory in Dimension Four

Table of Contents pages: 1 2

- Contents v6 free
- Preface xi12 free
- Chapter 0. Basic Notions and Notation 116 free
- Part 1. Classical Braids and Links 520
- Chapter 1. Braids 722
- Chapter 2. Braid Automorphisms 1934
- Chapter 3. Classical Links 2742
- 3.1. Knots and Links 2742
- 3.2. Basic Symmetries 2843
- 3.3. The Regular Neighborhood of a Link 2843
- 3.4. Trivial Links 2944
- 3.5. Split Union and Connected Sum 3045
- 3.6. Combinatorial Equivalence 3045
- 3.7. Regular Projections 3146
- 3.8. Link Diagrams 3247
- 3.9. Reidemeister Moves 3348
- 3.10. The Group of a Link 3550
- 3.11. A Note on Knots as Embeddings 3853

- Chapter 4. Braid Presentation of Links 4156
- Chapter 5. Deformation Chain and Markov's Theorem 4762

- Part 2. Surface Knots and Links 5166
- Part 3. Surface Braids 97112
- Chapter 13. Branched Coverings 99114
- Chapter 14. Surface Braids 105120
- Chapter 15. Products of Surface Braids 113128
- Chapter 16. Braided Surfaces 117132
- Chapter 17. Braid Monodromy 123138
- Chapter 18. Chart Descriptions 129144
- 18.1. Introduction 129144
- 18.2. BWTS Charts 129144
- 18.3. Enlarged BWTS Charts 132147
- 18.4. Surface Braid Charts 135150
- 18.5. Prom Charts to Surface Braids, I 135150
- 18.6. From Surface Braids to Charts 136151
- 18.7. From Charts to Surface Braids, II 138153
- 18.8. A Chart as the Singularity of a Projection 139154
- 18.9. From Charts to Braid Monodromies: Intersection Braid Word 140155
- 18.10. From Charts to Braid Systems 141156
- 18.11. Chart Moves 142157
- 18.12. Further Examples of Chart Moves 146161

- Chapter 19. Non-simple Surface Braids 149164
- Chapter 20. 1-Handle Surgery on Surface Braids 155170

- Part 4. Braid Presentation of Surface Links 157172
- Chapter 21. The Normal Braid Presentation 159174
- Chapter 22. Braiding Ribbon Surface Links 173188
- Chapter 23. Alexander's Theorem in Dimension Four 179194
- 23.1. Closed Surface Braids in D[sup(2)] x S[sup(2)] 179194
- 23.2. Closed Surface Braids in R[sup(4)] 180195
- 23.3. Alexander's Theorem in Dimension Four 181196
- 23.4. The Chart Description of a Surface Link 181196
- 23.5. The Braid Index of a Surface Link 182197
- 23.6. Another Kind of Braid Presentation 182197
- 23.7. Notes 182197

- Chapter 24. Split Union and Connected Sum 183198
- Chapter 25. Markov's Theorem in Dimension Four 187202
- Chapter 26. Proof of Markov's Theorem in Dimension Four 191206
- 26.1. Introduction 191206
- 26.2. Division of a Surface 191206
- 26.3. General Position with Respect to l 193208
- 26.4. ε Operation 193208
- 26.5. Deformation Chains 194209
- 26.6. Operations at the Division Level 195210
- 26.7. An Interpretation of Markov's Theorem in Dimension Four 196211
- 26.8. Proofs of Theorems 26.15 and 26.16 198213
- 26.9. Notation 198213
- 26.10. Existence of a Sawtooth, I 199214
- 26.11. Proof of the Sawtooth Lemma 202217
- 26.12. Mesh Division 202217
- 26.13. Existence of a Sawtooth, II 203218
- 26.14. Replacement of a Sawtooth 204219
- 26.15. Height Reduction, I 206221
- 26.16. Height Reduction, II 211226
- 26.17. Height Reduction, III 213228
- 26.18. Height Reduction, IV 216231
- 26.19. Height Reduction, V 219234
- 26.20. Proof of the Height Reduction Lemma II 221236
- 26.21. Proof of the Height Reduction Lemma I 222237

- Part 5. Surface Braids and Surface Links 223238

Table of Contents pages: 1 2