  We characterize the fractional Dehn twist coefficient (FDTC) on the
$n$-stranded braid group as the unique homogeneous quasimorphism to the real
numbers of defect at most 1 that equals 1 on the positive full twist and
vanishes on the $(n-1)$-stranded braid subgroup. In a different direction, we
establish that the slice-Bennequin inequality holds with the FDTC in place of
the writhe. In other words, we establish an affine linear lower bound for the
smooth slice genus of the closure of a braid in terms of the braid's FDTC. We
also discuss connections between these two seemingly unrelated results. In the
appendix we provide a unifying framework for the slice-Bennequin inequality and
its counterpart for the FDTC.
