The defect of Bennequin-Eliashberg inequality and Bennequin surfaces

For a null-homologous transverse link $\mathcal T$ in a general contact manifold with an open book, we explore strongly quasipositive braids and Bennequin surfaces. We define the defect $\delta(\mathcal T)$ of the Bennequin-Eliashberg inequality. We study relations between $\delta(\mathcal T)$ and minimal genus Bennequin surfaces of $\mathcal T$. In particular, in the disk open book case, under some large fractional Dehn twist coefficient assumption, we show that $\delta(\mathcal T)=N$ if and only if $\mathcal T$ is the boundary of a Bennequin surface with exactly $N$ negatively twisted bands. That is, the Bennequin inequality is sharp if and only if it is the closure of a strongly quasipositive braid.