  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.
