Local detection of strongly irreducible Heegaard splittings via knot exteriors

Let H1∪ΣH2 be a strongly irreducible Heegaard splitting of a 3-manifold M other than S3, and X a 3-dimensional submanifold of M such that: (1) X is homeomorphic to the exterior of a non-trivial knot in S3, and (2) there is a compressing disk, say DX, of ∂X such that ∂DX is a meridian curve of X. Suppose that ∂X∩Σ consists of a non-empty collection of simple closed curves which are essential in both ∂X and Σ. Then we show that: (1) the closure of some component of Σ⧹∂X is an annulus and is parallel to an annulus in ∂X, and (2) each component of Σ∩X is a (possibly boundary parallel) meridional annulus.