Trisections and link surgeries

We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby \cite{G-K3}. These links lie on Heegaard surfaces in $\#^j S^1 \times S^2$ and have surgeries yielding $\#^k S^1 \times S^2$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer.