Given two orientable, cusped hyperbolic 3-manifolds containing certain thrice-punctured spheres, Adams gave a diagrammatic definition for a third such manifold, their belted sum. Generalizing Adams' definition slightly, this work considers belted sum decompositions of fully augmented links (or FALs) in which all summands involved are also FALs. To do so, we provide explicit classifications of thrice-punctured spheres in FAL complements, making them easily recognizable. These classifications are used to characterize belted sum prime FALs geometrically, combinatorially and diagrammatically. Finally we prove that, in the context of belted sums, every FAL canonically decomposes into FALs which are either prime or two-fold covers of the Whitehead link.

This research was supported in part by NSF-REU Grants DMS-1461286 and DMS-1758020, as well as California State University, San Bernardino. We are very grateful to Christian Millichap for many (many!) conversations and suggestions that have significantly improved this work. Thanks also to Jeff Meyer for helpful conversations. Finally, thanks to the referee for their careful reading and helpful suggestions that improved this work.