The amalgamation of high distance Heegaard splittings is always efficient

Abstract

Let M be a compact orientable manifold, and F be an essential closed surface which cuts M into two 3-manifolds M1 and M2. Let \({M_{i}=V_{i}\cup_{S_{i}} W_{i}}\) be a Heegaard splitting for i = 1, 2. We denote by d(Si) the distance of \({V_{i}\cup_{S_{i}} W_{i}}\) . If d(S1), d(S2) ≥ 2(g(M1) + g(M2) − g(F)), then M has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of \({V_{1}\cup_{S_{1}} W_{1}}\) and \({V_{2}\cup_{S_{2}} W_{2}}\) .