Hyperbolicity of Alternating Links in Thickened Surfaces with Boundary

TL;DR

This paper characterizes when links in thickened surfaces with boundary are hyperbolic, providing conditions that identify hyperbolic links in various 3-manifold contexts, including handlebodies and fiber bundles.

Contribution

It offers simple criteria for hyperbolicity of links in thickened surfaces with boundary and extends these results to embeddings in ambient manifolds, enriching the understanding of hyperbolic links.

Findings

Provides conditions for hyperbolicity of links in $F imes I$

Identifies hyperbolic links in handlebodies and fiber bundles

Includes numerous examples of hyperbolic links

Abstract

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.