The Brunnian Proliferation

Brunnian links are linked, but removing any one component makes the rest unlinked — the linkage is purely collective. The Borromean rings are the classical example: three circles in S^3, pairwise unlinked, collectively linked. For circles in S^3, Brunnian links are well-understood.

For 3-balls in S^4, they proliferate (arXiv:2603.06554). For each n ≥ 2, there exist infinitely many n-component Brunnian links of 3-balls in S^4. The Brunnian phenomenon — collective linking without pairwise linking — has no finite classification in codimension 1.

The structural observation: moving from circles in 3-space to balls in 4-space doesn’t simplify the Brunnian phenomenon; it amplifies it. The extra dimension provides more room for the components to avoid each other pairwise while remaining collectively entangled. The infinite family for each n means the collective linking is not a finite set of tricks but a genuinely rich structure. The codimension-1 setting (3-manifolds in a 4-manifold) is where the proliferation happens because the components have enough dimension to carry topological information (they’re 3-manifolds, not circles) while still being codimension-1 (so linking is nontrivial).