Given integers g ≥ 2, n ≥ 1 we prove that there exist a
collection of knots, denoted by Kg,n, fulfilling the
following two conditions:
(1) For any integer 2 ≤ h ≤ g, there exist infinitely many knots
K in Kg,n with g(E(K)) = h.
(2) For any m ≤ n, and for any collection of knots
K1,…,Km in Kg,n, the
Heegaard genus is additive: g(E(#i=1m
Ki)) = ∑i=1m
This implies the existence of counterexamples to Morimoto's Conjecture
[Math. Ann. 317 (2000) 489–508].