Notes on the braid index of closed positive braids

The Morton–Franks–Williams inequality for a link gives a lower bound for the braid index in terms of the HOMFLY polynomial. Franks and Williams conjectured that for any closed positive braid the lower bound coincides with the braid index. In this paper, we show that the bound is achieved for a certain class of closed positive braids. We also give an infinite family of prime closed positive braids such that the lower bound does not coincide with their braid indices.