A classic theorem in 3D tells us that a split link of two components (i.e. two knots in the 3-sphere that don’t link each other at all) can be split apart in a unique way. On the contrary, in recent joint work I showed that some split links of surfaces (i.e. two surfaces in S^4 that don’t “link” each other at all) can be split apart in multiple ways. (Or in other words, there exist two non-isotopic splitting spheres in the complement of such a link.) In this talk, I’ll discuss the classic 3D situation, what goes wrong with extending this theorem analogously to dimension 4, and some interesting theorems about knots in 4D that contradict classical intuition from 3D. This is joint work with Mark Hughes (BYU) and Seungwon Kim (Sungkyunkwan).