We prove that if $K_1 \subset M_1,...,K_n \subset M_n$ are m-small knots in closed orientable 3-manifolds then the Heegaard genus of $E(#_{ i=1}^n K_i)$ is strictly less than the sum of the Heegaard genera of the $E(K_i)$ ($i=1,...,n$) if and only if there exists a proper subset $I$ of $\{ 1,...,n\}$ so that $#_{ i \in I} K_i$ admits a primitive meridian. This generalizes the main result of Morimoto in \cite{ morimoto1}.
Categories: math.GT