Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum - Li, Fengling; Lei, Fengchun; Yang, Guoqiu (Mathematica Scandinavica, 2014)
Let M_i be a compact orientable 3-manifold, and A_i an incompressible annulus on a component F_i of M_i, i=1,2. Suppose A_1 is separating on F_1 and A_2 is non-separating on F_2. Let M be the annulus sum of M_1 and M_2 along A_1 and A_2. In the present paper we show that if M_i has a Heegaard splitting V_i_S_i W_i with Heegaard distance d(S_i)≥2g(M_i)+5 for i=1,2, then g(M)=g(M_1)+g(M_2). Moreover, when g(F_2)≥ 2, the minimal Heegaard splitting of M is unique up to isotopy.