  We study the Morton-Franks-Williams inequality for closures of simple braids
(also known as positive permutation braids). This allows to prove, in a simple
way, that the set of simple braids is a orthonormal basis for the inner product
of the Hecke algebra of the braid group defined by K\'alm\'an, who first
obtained this result by using an interesting connection with Contact Topology.
  We also introduce a new technique to study the Homflypt polynomial for
closures of positive braids, namely resolution trees whose leaves are simple
braids. In terms of these simple resolution trees, we characterize closed
positive braids for which the Morton-Franks-Williams inequality is strict. In
particular, we determine explicitly the positive braid words on three strands
whose closures have braid index three.
