Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic

Abstract

Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks punctured twice by the alternating link were added. Lackenby proved that the first and second collections of links together form a closed subset of the set of all finite volume hyperbolic 3-manifolds in the geometric topology. Adams showed hyperbolicity for generalized augmented alternating links, which include additional trivial components that bound n-punctured disks for $n \geq 2$. Here we prove that generalized augmented cellular alternating links in I-bundles over closed surfaces are also hyperbolic and that in $S \times I$, the cellular alternating links and the augmented cellular alternating together form a closed subset of finite volume hyperbolic 3-manifolds in the geometric topology. Explicit examples of additional links in $S \times I$ to which these results apply are included.