On Heegaard splittings of knot exteriors with tunnel number degenerations

Let K1,K2 be two knots with t(K1)+t(K2)>2 and t(K1#K2)=2. Then, in the present paper, we will show that any genus three Heegaard splitting of E(K1#K2) is strongly irreducible and that E(K1#K2) has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a complete classification of those four genus three Heegaard splittings and describe unknotting tunnel systems of knots K1#K2 corresponding to those Heegaard splittings.