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.
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asked May 28, 2020 at 18:27
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