Sharpness of the Morton-Franks-Williams inequality for positive knots and links

We provide a combinatorial characterisation of positive diagrams satisfying the equality in the Morton-Franks-Williams bound for the degrees of the HOMFLY-PT polynomial. This characterisation allows generating with relative ease examples of diagrams realizing the crossing number, the braid index, and the maximal self-linking number. Besides, we suggest a conjecture concerning the sharpness of the Morton-Franks-Williams inequality for strongly quasipositive links.