Appendix 2 - Threading knot diagrams (by Hugh R. Morton), [Math. Proc. Camb. Phil. Soc. 99(1986), 247–260]

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.