#### Volume 8, issue 2 (2008)

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Knot exteriors with additive Heegaard genus and Morimoto's Conjecture

### Tsuyoshi Kobayashi and Yo’av Rieck

Algebraic & Geometric Topology 8 (2008) 953–969
##### Abstract

Given integers $g\ge 2$, $n\ge 1$ we prove that there exist a collection of knots, denoted by ${\mathsc{K}}_{g,n}$, fulfilling the following two conditions:

(1) For any integer $2\le h\le g$, there exist infinitely many knots $K\in {\mathsc{K}}_{g,n}$ with $g\left(E\left(K\right)\right)=h$.

(2) For any $m\le n$, and for any collection of knots ${K}_{1},\dots ,{K}_{m}\in {\mathsc{K}}_{g,n}$, the Heegaard genus is additive:

$g\left(E\left({#}_{i=1}^{m}{K}_{i}\right)\right)=\sum _{i=1}^{m}g\left(E\left({K}_{i}\right)\right).$

This implies the existence of counterexamples to Morimoto’s Conjecture [Math. Ann. 317 (2000) 489–508].

##### Keywords
Heegaard splitting, tunnel number, knot, composite knot
Primary: 57M25
Secondary: 57M27