The classification of certain linked 3-manifolds in 6-space

We classify smooth Brunnian (i.e., unknotted on both components) embeddings (S² × S¹) ⊔ S³ → ℝ⁶. Any Brunnian embedding (S² × S¹) ⊔ S³ → ℝ⁶ is isotopic to an explicitly constructed embedding f_k,m,n for some integers k, m, n such that m ≡ n (mod 2). Two embeddings f_k,m,n and f_{k′ ,m′,n′} are isotopic if and only if k = k′, m ≡ m′ (mod 2k) and n ≡ n′ (mod 2k). We use Haefliger’s classification of embeddings S³ ⊔ S³ → ℝ⁶ in our proof. The relation between the embeddings (S² × S¹) ⊔ S³ → ℝ⁶ and S³ ⊔ S³ → ℝ⁶ is not trivial, however. For example, we show that there exist embeddings f: (S² ×S¹) ⊔ S³ → ℝ⁶ and g, g′ : S³ ⊔ S³ → ℝ⁶ such that the componentwise embedded connected sum f # g is isotopic to f # g′ but g is not isotopic to g′.