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Braids and Self-Distributivity

  • Patrick Dehornoy

Part of the Progress in Mathematics book series (PM, volume 192)

Table of contents

  1. Front Matter
    Pages N1-xix
  2. Ordering the Braids

    1. Front Matter
      Pages 1-1
    2. Patrick Dehornoy
      Pages 3-46
    3. Patrick Dehornoy
      Pages 47-96
    4. Patrick Dehornoy
      Pages 97-140
    5. Patrick Dehornoy
      Pages 141-174
    6. Patrick Dehornoy
      Pages 177-233
    7. Patrick Dehornoy
      Pages 235-283
  3. Free LD-systems

    1. Front Matter
      Pages 175-175
    2. Patrick Dehornoy
      Pages 285-330
    3. Patrick Dehornoy
      Pages 331-384
    4. Patrick Dehornoy
      Pages 385-442
  4. Other LD-Systems

    1. Front Matter
      Pages 443-443
    2. Patrick Dehornoy
      Pages 445-487
    3. Patrick Dehornoy
      Pages 489-535
    4. Patrick Dehornoy
      Pages 537-570
    5. Patrick Dehornoy
      Pages 571-601
  5. Back Matter
    Pages 603-623

About this book

Introduction

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.

Keywords

Group theory Gruppentheorie Knot theory algebra algebraische Geomentrie topology

Authors and affiliations

  • Patrick Dehornoy
    • 1
  1. 1.Laboratoire SDAD MathématiquesUniversité Campus 2CaenFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-8442-6
  • Copyright Information Birkhäuser Verlag 2000
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-9568-2
  • Online ISBN 978-3-0348-8442-6
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site