We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting handlebodies has positive genus, and gluing a pair of pieces together along the twice-punctured disks in their boundaries, we show the result is also hyperbolic. This should be contrasted with composition of any pair of knots in the 3-sphere, which is never hyperbolic. Similar results are obtained when both twice-punctured disks are in the same handlebody and we glue a resultant piece to itself along copies of the twice-punctured disks on its boundary. We include applications to staked links.

This research was supported by Williams College and NSF Grant DMS-1947438. A special thanks to the other members of the knot theory research group in the SMALL REU program at Williams College in summer of 2022, including Alexandra Bonat, Maya Chande, Maxwell Jiang, Zachary Romrell, Benjamin Shapiro and Dora Woodruff. Without the many helpful conversations with them, this paper would not have come into being. And thanks also to the referee for helpful coments.