Tangles and tubing operations

Let (B, T) be an n string tangle, E(T) the exterior cl(B − N(T)) and P the punctured sphere cl(∂B − N(T)). The tangle (B, T) is called atomic if it does not contain a nonsplit tangle with k < n essentially. For a string s of T the surface T(s) = P∪(E(T)∩N(s)) is said to be obtained by performing a tubing operation on P along s. We give a necessary and sufficient condition for T(s) to be incompressible in E(T), when (B, T) is atomic. We show also that if a knot K is decomposed into two atomic tangles with no parallel pairs of strings, then every nontrivial Dehn surgery on K yields a laminar manifold.