Unlike classical grasp synthesis, where force-closure grasps are determined based on local contact information, the caging condition requires reasoning about connected components of the free configuration space of an object. In our work, we propose a novel framework for the synthesis and provably correct verification of caging grasps on a class of 3D objects which exhibit geometric features we call necks and double forks. We consider the synthesis and verification of a caging grasps on such objects by means of one or more caging tools that can be arranged to form an approximate closed loop, a caging loop, around the identified neck or double fork features of the object. As a key theoretical tool, we utilize an augmentation of the classical notion of the linking number from algebraic topology allowing us to prove sufficient conditions for a cage to exist even in the case when the object under consideration is partially deformable. We present both our formalization of caging with caging loops as well as an algorithm and evaluation for synthesizing and verifying such caging grasps on triangular surface meshes in 3D.