  Positive permutation braids on n strings, which are defined to be positive
n-braids where each pair of strings crosses at most once, form the elementary
but non-trivial building blocks in many studies of conjugacy in the braid
groups. We consider conjugacy among these elementary braids which close to
knots, and show that those which close to the trivial knot or to the trefoil
are all conjugate. All such n-braids with the maximum possible crossing number
are also shown to be conjugate.
  We note that conjugacy of these braids for n<6 depends only on the crossing
number. In contrast, we exhibit two such braids on 6 strings with 9 crossings
which are not conjugate but whose closures are each isotopic to the (2,5) torus
knot.
