Abstract
Let H 1∪ Σ H 2 be a strongly irreducible Heegaard splitting of a 3-manifold M other than S 3, and X a 3-dimensional submanifold of M such that: (1) X is homeomorphic to the exterior of a non-trivial knot in S 3, and (2) there is a compressing disk, say D X , of ∂X such that ∂D X 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.
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