I recently heard a beautiful talk by Yuliy Baryshnikov on the general question of when an object can be pinned by some set of fixed points. They consider arbitrary objects in 2D and prove the following theorem:

Let D be a planar domain. Either one can pull a configuration C of two points $\{p_1,p_2\}$ around D, or there exists a full rotation of C entirely within D, that is a loop π′: $S^1$ → E (E being the Euclidean group of transformations) such that the vector $π′\circ p_1 − π′ \circ p_2$ turns around the origin (perhaps, several times).

They use a topological approach which uses Mayer-Vietoris sequences in homology; apparently to generalize to 3D one must use Mayer-Vietoris spectral sequences, though this is "future work".

The slides are here and does include some discussion of computing the possibility of caging / linking effectively, but again, they focus on the 2D problem.

I recently heard a beautiful talk by Yuliy Baryshnikov on the general question of when an object can be pinned by some set of fixed points. They consider arbitrary objects in 2D and prove the following theorem:

Let D be a planar domain. Either one can pull a configuration C of two points $\{p_1,p_2\}$ around D, or there exists a full rotation of C entirely within D, that is a loop π′: $S^1$ → E (E being the Euclidean group of transformations) such that the vector $π′\circ p_1 − π′ \circ p_2$ turns around the origin (perhaps, several times).

They use a topological approach which uses Mayer-Vietoris sequences in homology; apparently to generalize to 3D one must use Mayer-Vietoris spectral sequences, though this is "future work".

The slides are here and do include some discussion of computing the possibility of caging / linking effectively, but again, they focus on the 2D problem.