Morimoto's Conjecture for m-small knots
  Let $X$ be the exterior of connected sum of knots and $X_i$ the exteriors of the individual knots. In \cite{morimoto1} Morimoto conjectured (originally for $n=2$) that $g(X) < \sigma_{i=1}^n g(X_i)$ if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of the knots. For m-small knots we prove this conjecture and bound the possible degeneration of the Heegaard genus (this bound was previously achieved by Morimoto under a weak assumption \cite{morimoto2}):   $$\sigma_{i=1}^n g(X_i) - (n-1) \leq g(X) \leq \sigma_{i=1}^n g(X_i).$$ 
Tsuyoshi Kobayashi and Yo'av Rieck