Heegaard genus of the connected sum of m-small knots

摘要

We prove that if K-1 subset of M-1,..., K-n subset of M-n are m-small knots in closed orientable 3-manifolds, then the Heegaard genus of E(#K-n(i=1)i) is strictly less than the sum of the Heegaard genera of the E (K-i) (i = 1,..., n) if and only if there exists a proper subset I of {1,...,n} so that #K-i is an element of I(i) admits a primitive meridian. This generalizes the main result of Morimoto [On the super additivity of tunnel number of knots, Math. Ann. 317 (2000), 489-508].