Stabilized Heegaard Splitting Of Complements Of Composite Knots

Let K be a knot in S3, E(K) is the complement of K in S3, V ∪s W is a Heegaard splitting of E(K). T=(?)E(K) (?)_W, and A is a meridional annulus on T. If there is an essential disk B in V, and a planar surface P in W. satisfying1. One boundary component of P lies in S, denoted by α, the other boundary components of P lie in A,2. B intersects P in one point, and3. There exists a fiber arc γ in W, let N(γ) be a regular neighborhood of γ in W, such that b. N(γ)∩ P as in figure 1. We denote n=|(?)P|-1, then we call V ∪s W is n-stabilized. In this paper we prove the following theorem:Let K1 and K2 be two knots in S3, Vi ∪si Wi is a Heegaard splitting of E(Ki), and (?)E(Ki) (?)_Wi, for i=1,2. Then the dual amalgamation of Vi ∪S1 W1 and V2 ∪ s2 W2 is a Heegaard splitting of E(K1K2). If one of V1∪s1 Wi and V2 ∪s2 W2 is n-stabilized, then the dual amalgamation of them is stabilized.