Table of contents for Ordering braids / Patrick Dehornoy ; with Ivan Dynnikov, Dale Rolfsen, Bert Wiest.


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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 a-ordering of B,                                             19
2. Local properties of the a-ordering                               26
3. Global properties of the a-ordering                              29
4. The a-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
3. The increasing enumeration of Div(Ad)                           117
Chapter VII. Alternating Decompositions                              129
1. The 4nF-splitting of a braid in B+                              130
2. The 4-normal form                                               135
3. Burckel's approach                                              143
4. Applications                                                    148
Chapter VIII. Dual Braid Monoids                                     153
1. Dual braid monoids                                              154
2. The 0-normal form on B'*                                        159
3. Connection between orders                                       163
Chapter IX. Automorphisms of a Free Group                            173
1. Artin representation of a-positive braids                       173
2. From an automorphism back to a braid                            178
3. Pulling back orderings of free groups                           182
Chapter X. Curve Diagrams                                            185
1. A braid ordering using curve diagrams                           185
2. Proof of Properties A, C, and S                                 189
Chapter XI. Relaxation Algorithms                                    195
1. Bressaud's regular language of relaxation braids                196
2. The transmission-relaxation normal form of braids               204
Chapter XII. Triangulations                                          221
1. The coordinates of a braid                                      222
2. Triangulations and laminations                                  225
3. The Mosher normal form                                          236
Chapter XIII. Hyperbolic Geometry                                    247
1. Uncountably many orderings of the braid group                   248
2. The classification of orderings induced by the action on R      256
3. The subword property for all Nielsen-Thurston type orderings    263
Chapter XIV. The Space of all Braid Orderings                        265
1. The spaces of orderings on a group                              265
2. The space of left-orderings of the braid groups                 268
Chapter XV. Bi-ordering the Pure Braid Groups                        273
1. Lower central series                                            273
2. Artin coordinates and Magnus expansion                          274
3. The Magnus ordering of PB,                                      279
4. The ordering of positive pure braids                            283
5. Incompatibility of the orderings                                286
Chapter XVI. Open Questions and Extensions                           291
1. General questions                                               291
2. More specific questions                                         293
3. Generalizations and extensions                                  301
Key Definitions                                                   309
Bibliography                                                      311
Index of Notation                                                 319



Library of Congress subject headings for this publication: Braid theory, Linear orderings