Morimoto's Conjecture for m-small knots

Details Morimoto's Conjecture for m-small knots Morimoto's Conjecture for m-small knots

2002-12-27

arxiv.org/abs/math/0212349v1

Mathematics

Geometric Topology

17 pages; to appear in the proceedings of the conference "Musubime no topology (Topology of knots) V" held at Waseda Universit

Scientific paper

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}): $$\sigma_{i=1}^n g(X_i) - (n-1) \leq g(X) \leq \sigma_{i=1}^n g(X_i).$$

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