  Our main result is a version of Birman's theorem about equivalence of plats,
which does not involve stabilization, for the unlink. We introduce the pocket
and flip moves, which modify a plat without changing its link type or bridge
index. Theorem 1 shows that using the pocket and flip moves, one can simplify
any closed $n$-plat presentation of the unknot to the standard 0-crossing
diagram of the unknot, through a sequence of plats of non-increasing bridge
index. The theorem readily generalises to the case of the unlink.
