Book contents
- Frontmatter
- Preface
- Contents
- Chapter I BRAIDS AND CONFIGURATION SPACES
- Chapter II BRAIDS AND LINKS
- Chapter III POLYNOMIAL COVERING MAPS
- Chapter IV ALGEBRA AND TOPOLOGY OF WEIERSTRASS POLYNOMIALS
- Appendix 1 A presentation for the abstract coloured braid group (by Lars Gæde)
- Appendix 2 Threading knot diagrams (by Hugh R. Morton), [Math. Proc. Camb. Phil. Soc. 99(1986), 247–260]
- Bibliography
- Index (to Chapters I–IV and Appendix 1)
Appendix 2 - Threading knot diagrams (by Hugh R. Morton), [Math. Proc. Camb. Phil. Soc. 99(1986), 247–260]
Published online by Cambridge University Press: 06 January 2010
- Frontmatter
- Preface
- Contents
- Chapter I BRAIDS AND CONFIGURATION SPACES
- Chapter II BRAIDS AND LINKS
- Chapter III POLYNOMIAL COVERING MAPS
- Chapter IV ALGEBRA AND TOPOLOGY OF WEIERSTRASS POLYNOMIALS
- Appendix 1 A presentation for the abstract coloured braid group (by Lars Gæde)
- Appendix 2 Threading knot diagrams (by Hugh R. Morton), [Math. Proc. Camb. Phil. Soc. 99(1986), 247–260]
- Bibliography
- Index (to Chapters I–IV and Appendix 1)
Summary
Introduction
Alexander showed that an oriented link K in S3 can always be represented as a closed braid. Later Markov-described (without full details) how any two such representations of K are related. In her book, Birman gives an extensive description, with a detailed combinatorial proof of both these results.
In this paper I shall describe a simple method of representing an oriented link K as a closed braid, starting from a knot diagram for K and ‘threading’ a suitable unknotted curve L through the strings of K so that K is braided relative to L, i.e. K ∪ L forms a closed braid together with its axis.
I shall then give a straightforward derivation of Markov's result, using the ideas of threading, and a geometric version of the braid moves with which Markov relates two braids representing the same K. The geometric approach is described in terms of links K ∪ L, in which K forms a closed braid relative to an axis L. Such a link will be called braided, and in addition it will be called a threading of an explicit diagram for K if it arises from the threading construction. Two braided links which are related by the geometric version of Markov's moves will be called Markov-equivalent.
- Type
- Chapter
- Information
- Braids and CoveringsSelected Topics, pp. 171 - 184Publisher: Cambridge University PressPrint publication year: 1989
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