Codimension one embeddings of product of three spheres

Let f :Sp×Sq×Sr→Sp+q+r+1 be a smooth embedding with 1⩽p⩽q⩽r. For p⩾2, the authors have shown that if p+q≠r, or p+q=r and r is even, then the closure of one of the two components of Sp+q+r+1−f(Sp×Sq×Sr) is diffeomorphic to the product of two spheres and a disk, and that otherwise, there are infinitely many embeddings, called exotic embeddings, which do not satisfy such a property. In this paper, we study the case p=1 and construct infinitely many exotic embeddings. We also give a positive result under certain (co)homological hypotheses on the complement. Furthermore, we study the case (p,q,r)=(1,1,1) more in detail and show that the closures of the two components of S4−f(S1×S1×S1) are homeomorphic to the exterior of an embedded solid torus or Montesinosʹ twin in S4.