Morimoto's Conjecture for m-small knots

Abstract

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}): $${\sigma }_{i=1}^{n}g(X_i)-(n-1)\le g(X)\le {\sigma }_{i=1}^{n}g(X_i).$$

17 pages; to appear in the proceedings of the conference "Musubime no topology (Topology of knots) V" held at Waseda University, 16-19 December, 2002