Hyperbolicity of Alternating Links in Thickened Surfaces with Boundary

Let $F$ be a compact orientable surface with nonempty boundary other than a disk. Let $L$ be a link in $F\times I$ with a connected weakly prime cellular alternating projection to $F$. We provide simple conditions that determine exactly when $(F\times I)\setminus N(L)$ is hyperbolic. We also consider suitable embeddings of $F\times I$ in an ambient manifold $Y$ with boundary and provide conditions on links $L\subset F\times I$ which guarantee tg-hyperbolicity of $Y\setminus N(L)$. These results provide many examples of hyperbolic links in handlebodies and fiber bundles. It also provides many examples of staked links that are hyperbolic.