The Slice-Bennequin Inequality for the Fractional Dehn Twist Coefficient

We characterize the fractional Dehn twist coefficient (FDTC) on the n -stranded braid group as the unique homogeneous quasimorphism to R 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.