Contents
Preface ix
Introduction 1
A meeting of two classical subjects 1
A convergence of approaches 3
Organization of the text 5
Guidelines to the reader 7
Acknowledgements 8
Chapter I. Braid Groups 9
1. The Artin presentation 9
2. Isotopy classes of braid diagrams 10
3. Mapping class groups 12
4. Positive braids 14
Chapter II. A Linear Ordering of Braids 19
1. The σ-ordering of Bn 19
2. Local properties of the σ-ordering 26
3. Global properties of the σ-ordering 29
4. The σ-ordering of positive braids 35
Chapter III. Applications of the Braid Ordering 43
1. Consequences of orderability 44
2. Applications of more specific properties 46
3. Application of well-orderability 50
Chapter IV. Self-distributivity 55
1. Colouring positive braids 56
2. Colouring arbitrary braids 66
3. The group of left self-distributivity 76
4. Normal forms in free LD-systems 81
5. Appendix: Iterations of elementary embeddings in set theory 84
Chapter V. Handle Reduction 87
1. Description of handle reduction 87
2. Convergence of handle reduction 92
3. Special cases and variants 102
Chapter VI. Connection with the Garside Structure 107
1. The degree of a positive braid 108
2. Proving Property C using a counting argument 113
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