Symplectic $2$-handles and transverse links

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching $2$-handles to convex boundaries of symplectic $4$-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over $S^1$; in this case the handles can be attached with any framing larger than a lower bound determined by the fibration. The other method results in a weaker convexity condition on the new boundary (sufficient to imply tightness of the new contact structure), and in this case the handles can be attached with any framing less than a certain upper bound. These methods supplement methods developed by Weinstein and Eliashberg for attaching symplectic $2$-handles along Legendrian knots.