Heegaard genus of the connected sum of m-small knots

Abstract

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}.