Thin position of a pair (3–manifold

We introduce the notions of Heegaard splittings and thin multiple Heegaard splittings of 1-submanifolds in compact orientable 3-manifolds, which are generalizations of those of bridge decompositions and thin positions. We show that ei-ther a thin multiple Heegaard splitting of 1-submanifold T is also a Heegaard splitting with minimal complexity or the exterior of T contains an essential surface with meridional boundary other than the boundary parallel annulus. 1. Introduction. The notion of thin position for knots in S3 is introduced by D. Gabai in [G], and played important roles in solutions of the property R conjecture by Gabai and the knot complement conjecture by C.McA. Gordon and J. Luecke [G-L]. A. Thompson showed in [T] that if an exterior of a knot