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

The slice-Bennequin inequality for the fractional Dehn twist coefficient