# Generalized Brunnian links

A Brunnian link of order $$n$$ is nontrivial link of $$n$$ rings that becomes a trivial link of $$n-1$$ rings if any ring is removed. They were classified up to link-homotopy by Milnor in 1954. This suggests the following generalization. Let $$\mathcal{A}$$ be an antichain of subsets of $$\{1,2,\dots,n\}$$. In other words, $$\mathcal{A}$$ is a collection of subsets of $$\{1,2,\dots,n\}$$ such that if $$A,B\in\mathcal{A}$$ and $$A\subseteq B$$, then $$A=B$$. An $$\mathcal{A}$$-link is a link of $$n$$ rings $$C_1,\dots,C_n$$ such that $$\{C_{i_1},\dots,C_{i_j}\}$$ is a minimal set of rings whose removal separates all the remaining rings if and only if $$\{i_1,\dots,i_j\} \in\mathcal{A}$$. What can be said about $$\mathcal{A}$$-links? For what $$\mathcal{A}$$ do they exist? For instance, they obviously don't exist if $$\mathcal{A}$$ consists of the single set $$\{1,2,\dots,n\}$$. Can they be classified analogously to Milnor's result? There is a special case here.