We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1 \times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives.
"Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold." J. Differential Geom. 101 (1) 67 - 157, September 2015. https://doi.org/10.4310/jdg/1433975484