Knot exteriors with additive Heegaard genus and Morimoto's Conjecture

Abstract

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

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

(2) For any $m\le n$, and for any collection of knots $K_1,\dots ,K_m\in {\mathcal{K}}_{g,n}$, 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].