The defect of Bennequin-Eliashberg inequality and Bennequin surfaces

Abstract

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.

31 pages, 13 figures

References in corpus (3)

• Quasipositive links and Stein surfaces
• On a question of Etnyre and Van Horn-Morris
• Quasi right-veering braids and non-loose links

Cited by in corpus (1)

• Quasi right-veering braids and non-loose links