A lower bound of genus of amalgamations of Heegaard splittings.

In the paper, we give a lower bound on the genus of an amalgamation of two Heegaard splittings V1, ⋃S1, W1 for M1 and V2, ⋃S2, W2 for M2 along boundary components F1 and F2 under some conditions on the distances of the factor Heegaard splittings. A direct consequence is that under some circumstances, g(M) = g(M1) + g(M2) - g(F) holds, and another is that under some circumstances we have a better lower bound. [ABSTRACT FROM AUTHOR]