Open book embeddings of 3-manifolds after link sum with S1×S2 along links with two components

Open book embedding of closed orientable [Formula: see text]-manifolds into an open book of the [Formula: see text]-sphere [Formula: see text] is an open problem. In this paper, we discuss open book embeddings of [Formula: see text]-manifolds into the trivial open book of [Formula: see text]. More precisely, we show that given a closed orientable [Formula: see text]-manifold M, there exist two disjoint knots [Formula: see text] and [Formula: see text] in M such that the manifold [Formula: see text] obtained from M by taking a link sum of M and [Formula: see text] along [Formula: see text] and [Formula: see text] admits an open book which open book embeds into the trivial open book of [Formula: see text] where S and N are the south and north poles of [Formula: see text], respectively.