The slice-Bennequin inequality gives an upper bound for the self-linking number of a knot in terms of its four-ball genus. The s-Bennequin and τ-Bennequin inequalities provide upper bounds on the self-linking number of a knot in terms of the Rasmussen s invariant and the Ozsváth-Szabó τ invariant. We exhibit examples in which the difference between self-linking number and four-ball genus grows arbitrarily large, whereas the s-Bennequin inequality and the τ-Bennequin inequality are both sharp.

EA was partially supported by the Ford Foundation. KK was partially supported by Simons Foundation Collaboration Grants for Mathematicians and NSF grant DMS-2005450. LT was partially supported by NSF grants DMS-200553 and DMS-2104309.