The classification of certain linked 3-manifolds in 6-space (Q2803005)

\(S^1 \sqcup S^1\) embedded in \(\mathbb R^3\) is usually called a 1-link and \(S^p \sqcup S^p \to \mathbb R^q\) is a \(p\)-link in \(\mathbb R^q\). A very hard general question is the isotopy classification of links, so one puts additional conditions on the embeddings, for example that the embeddings are unknotted in both components; these are called Brunnian embeddings. A. Haefliger obtained a classification of \(S^3 \sqcup S^3 \to S^6\). The present paper treats smooth Brunnian embeddings of \((S^2 \times S^1) \sqcup S^3 \to S^6\) and simultaneously \(S^3 \sqcup S^3 \to S^6\). In this situation a Brunnian embedding is one that is isotopic to a standard one. Therefore, the paper brings explicit constructions of standard embeddings, denoted \(f_{k,m,n}\) in the \((S^2 \times S^1) \sqcup S^3 \to S^6\) case and and \(g_{k,m,n}\) in the case \(S^3 \sqcup S^3 \to S^6\), where \(k,m,n\) are integers such that \(m \equiv n\) (mod 2). Two standard embeddings \(f_{k,m,n}\) and \(f_{k',m',n'}\) are isotopic iff \(k = k', m \equiv m'\) mod (2\(k\)) and \(n \equiv n'\) mod (2\(k\)).NEWLINENEWLINEResults: 1. Any Brunnian embedding \(f: (S^2 \times S^1) \sqcup S^3 \to S^6\) is isotopic to some \(f_{k,m,n}\) such that \(m \equiv n\) mod (2) and two such embeddings are isotopic iff \(k=k'\), \(m \equiv m'\) mod (2\(k\)) and \(n \equiv n'\) mod (2\(k\)). 2. Any Brunnian embedding \(g: S^3\sqcup S^3 \to S^6\) is isotopic to some \(g_{m,n}\) such that \(m \equiv n\) mod (2) and two such embeddings are isotopic iff \(m = m'\) and \(n = n'\).