Studying links via closed braids. V: The unlink

The main result is a version of Markov’s Theorem which does not involve stabilization, in the special case of the r-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most 1. To state the result, we need the concept of an “exchange move”, which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem 1 shows that exchange moves are the only obstruction to reducing a closed n-braid representative of the r-component unlink to the standard closed r-braid representative, through a sequence of braids of nonincreasing braid index.