My first paper is about proving a version of Birman’s theorem about equivalence of plats (as knot types), which does not involve stabilisation, for the unlink. Joan Birman and William Menasco published a series of papers, in the 90’s, seeking to understand closed braid representatives of a link. They began by investigating the role of stabilisation in simplifying closed braids. In 1983, Hugh Morton had given an example of a 4-braid representative of the unknot which can not be simplified without first increasing the string index to 5. Menasco and Birman’s work began by trying to understand why stabilisation is necessary in Morton’s example. Morton’s example is crucial because it shows that not only is stabilisation an essential part of Markov’s theorem but also that it is an all pervasive phenomenon, by connect summing his example to any closed braid. Menasco and Birman discovered a new move, which they called the ‘exchange move’ and they proved that this move was the only obstruction to simplifying the n component unlink to its standard diagram, through a sequence of closed braids of non - increasing string index. They went on to generalise this result to all link types. Their work has inpired the main result of this paper, which is the first in a series of papers investigating plat representations of knots.
