Braids and Self-Distributivity

The aim of this book is to present recently discovered connections between Artin's braid groups En and left self-distributive systems (also called LD systems), which are sets equipped with a binary operation satisfying the left self-distributivity identity x(yz) = (xy)(xz). (LD) Such connections appeared in set theory in the 1980s and led to the discovery in 1991 of a left invariant linear order on the braid groups. Braids and self-distributivity have been studied for a long time. Braid groups were introduced in the 1930s by E. Artin, and they have played an increas ing role in mathematics in view of their connection with many fields, such as knot theory, algebraic combinatorics, quantum groups and the Yang-Baxter equation, etc. LD-systems have also been considered for several decades: early examples are mentioned in the beginning of the 20th century, and the first general results can be traced back to Belousov in the 1960s. The existence of a connection between braids and left self-distributivity has been observed and used in low dimensional topology for more than twenty years, in particular in work by Joyce, Brieskorn, Kauffman and their students. Brieskorn mentions that the connection is already implicit in (Hurwitz 1891). The results we shall concentrate on here rely on a new approach developed in the late 1980s and originating from set theory.

InhaltsangabeA: Ordering the Braids.- I. Braids vs. Self-Distributive Systems.- I.1 Braid Groups.- I.2 Braid Colourings.- I.3 A Self-Distributive Operation on Braids.- I.4 Extended Braids.- I.5 Notes.- II. Word Reversing.- II.1 Complemented Presentations.- II.2 Coherent Complements.- II.3 Garside Elements.- II.4 The Case of Braids.- II.5 Double Reversing.- II.6 Notes.- III. The Braid Order.- III.1 More about Braid Colourings.- III.2 The Linear Ordering of Braids.- III.3 Handle Reduction.- III.4 Alternative Definitions.- III.5 Notes.- IV. The Order on Positive Braids.- IV.1 The Case of Three Strands.- IV.2 The General Case.- IV.3 Applications.- IV.4 Notes.- IV.Appendix: Rank Tables.- B: Free LD-systems.- V. Orders on Free LD-systems.- V.1 Free Systems.- V.2 The Absorption Property.- V.3 The Confluence Property.- V.4 The Comparison Property.- V.5 Canonical Orders.- V.6 Applications.- V.7 Notes.- VI. Normal Forms.- VI.1 Terms as Trees.- VI.2 The Cuts of a Term.- VI.3 The O-Normal Form.- VI.4 The Right Normal Form.- VI.5 Applications.- VI.6 Notes.- VI.Appendix: The First Codes.- VII. The Geometry Monoid.- VII.1 Left Self-Distributivity Operators.- VII.2 Relations in the Geometry Monoid.- VII.3 Syntactic Proofs.- VII.4 Cuts and Collapsing.- VII.5 Notes.- VIII. The Group of Left Self-Distributivity.- VIII.1 The Group GLD and the Monoid MLD.- VIII.2 The Blueprint of a Term.- VIII.3 Order Properties in GLD.- VIII.4 Parabolic Subgroups.- VIII.5 Simple Elements in MLD.- VIII.6 Notes.- IX. Progressive Expansions.- IX.1 The Polish Algorithm.- IX.2 The Content of a Term.- IX.3 Perfect Terms.- IX.4 Convergence Results.- IX.5 Effective Decompositions.- IX.6 The Embedding Conjecture.- IX.7 Notes.- IX.Appendix: Other Algebraic Identities.- C: Other LD-Systems.- X. More LD-Systems.- X.1 The Laver Tables.- X.2 Finite Monogenic LD-systems.- X.3 Multi-LD-Systems.- X.4 Idempotent LD-systems.- X.5 Two-Sided Self-Distributivity.- X.6 Notes.- X.Appendix: The First Laver Tables.- XI. LD-Monoids.- XI.1 LD-Monoids.- XI.2 Completion of an LD-System.- XI.3 Free LD-Monoids.- XI.4 Orders on Free LD-Monoids.- XI.5 Extended Braids.- XI.6 Notes.- XII. Elementary Embeddings.- XII.1 Large Cardinals.- XII.2 Elementary Embeddings.- XII.3 Operations on Elementary Embeddings.- XII.4 Finite Quotients.- XII.5 Notes.- XIII. More about the Laver Tables.- XIII.1 A Dictionary.- XIII.2 Computation of Rows.- XIII.3 Fast Growing Functions.- XIII.4 Lower Bounds.- XIII.5 Notes.- List of Symbols.

A: Ordering the Braids.- I. Braids vs. Self-Distributive Systems.- II. Word Reversing.- III. The Braid Order.- IV. The Order on Positive Braids.- B: Free LD-systems.- V. Orders on Free LD-systems.- VI. Normal Forms.- VII. The Geometry Monoid.- VIII. The Group of Left Self-Distributivity.- IX. Progressive Expansions.- C: Other LD-Systems.- X. More LD-Systems.- XI. LD-Monoids.- XII. Elementary Embeddings.- XIII. More about the Laver Tables.- List of Symbols.