A note on generalized Thurston--Bennequin inequalities

We give a generalized Thurston–Bennequin-type inequality for links in $S^3$ using a Bauer–Furuta-type invariant for 4-manifolds with contact boundary. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth 4-manifold whose non-equivariant Bauer–Furuta invariant is non-zero.