Scharlemann proved that any 2-knot in the 4-sphere with four critical points must be unknotted. Four critical points is the minimum for any embedding — two minima and two maxima in a Morse-theoretic decomposition. In dimension 4, this minimum complexity forces triviality. A knot simple enough to have only four critical points cannot be genuinely knotted.
Kim, Nahm, and Tatsuoka show the opposite holds one dimension up. Using barbell diffeomorphisms, they construct infinitely many non-isotopic 3-knots in the 5-sphere, each with exactly four critical points. Same complexity measure, same minimum value, but the constraint that forces unknotting in dimension 4 permits infinite knotting in dimension 5. The construction also produces knotted solid tori in the 4-sphere and the 5-ball, resolving a conjecture by Budney and Gabai.
The structural point is about what dimension does to constraints. Four critical points is a topological budget — the simplest possible embedding. In dimension 4, this budget is too tight for knotting; all the available complexity is consumed by the embedding itself. In dimension 5, the same budget leaves room. The extra dimension provides enough geometric freedom that the minimum-complexity embedding can still support genuinely distinct topological types.
This is not a story about higher dimensions being “more complex” in some vague sense. It is a precise inversion: a theorem that holds in dimension n fails in dimension n+1, not because the objects are more complicated but because the space they live in provides different constraints. The same structural poverty — four critical points — means triviality in one dimension and infinite variety in the next. The boundary between dimensions is not a gradient. It is a threshold where qualitative behavior reverses.