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A Study of Braids

  • Kunio Murasugi
  • Bohdan I. Kurpita

Part of the Mathematics and Its Applications book series (MAIA, volume 484)

Table of contents

  1. Front Matter
    Pages i-x
  2. Kunio Murasugi, Bohdan I. Kurpita
    Pages 1-10
  3. Kunio Murasugi, Bohdan I. Kurpita
    Pages 11-30
  4. Kunio Murasugi, Bohdan I. Kurpita
    Pages 31-56
  5. Kunio Murasugi, Bohdan I. Kurpita
    Pages 57-72
  6. Kunio Murasugi, Bohdan I. Kurpita
    Pages 74-95
  7. Kunio Murasugi, Bohdan I. Kurpita
    Pages 96-112
  8. Kunio Murasugi, Bohdan I. Kurpita
    Pages 113-127
  9. Kunio Murasugi, Bohdan I. Kurpita
    Pages 128-145
  10. Kunio Murasugi, Bohdan I. Kurpita
    Pages 146-166
  11. Kunio Murasugi, Bohdan I. Kurpita
    Pages 167-189
  12. Kunio Murasugi, Bohdan I. Kurpita
    Pages 190-213
  13. Kunio Murasugi, Bohdan I. Kurpita
    Pages 214-218
  14. Back Matter
    Pages 219-277

About this book

Introduction

In Chapter 6, we describe the concept of braid equivalence from the topological point of view. This will lead us to a new concept braid homotopy that is discussed fully in the next chapter. As just mentioned, in Chapter 7, we shall discuss the difference between braid equivalence and braid homotopy. Also in this chapter, we define a homotopy braid invariant that turns out to be the so-called Milnor number. Chapter 8 is a quick review of knot theory, including Alexander's theorem. While, Chapters 9 is devoted to Markov's theorem, which allows the application of this theory to other fields. This was one of the motivations Artin had in mind when he began studying braid theory. In Chapter 10, we discuss the primary applications of braid theory to knot theory, including the introduction of the most important invariants of knot theory, the Alexander polynomial and the Jones polynomial. In Chapter 11, motivated by Dirac's string problem, the ordinary braid group is generalized to the braid groups of various surfaces. We discuss these groups from an intuitive and diagrammatic point of view. In the last short chapter 12, we present without proof one theorem, due to Gorin and Lin [GoL] , that is a surprising application of braid theory to the theory of algebraic equations.

Keywords

Group theory Homotopy Mathematica algebra mathematics

Authors and affiliations

  • Kunio Murasugi
    • 1
  • Bohdan I. Kurpita
    • 2
  1. 1.University of TorontoTorontoCanada
  2. 2.The Daiwa Anglo-Japanese FoundationTorontoCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-9319-9
  • Copyright Information Springer Science+Business Media B.V. 1999
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5245-2
  • Online ISBN 978-94-015-9319-9
  • Buy this book on publisher's site