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Given integers g_i > 1 (i=1,...,n) we prove that there exist infinitely mayknots K_i in S^3 so that g(E(K_i)) = g_i and the Heegaard genus of the exteriorof the connected sum of K_1,...,K_n is the sum the Heegaard genera ofK_1,...,K_n, that is: g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). (Here, E()denotes the exterior and g() the Heegaard genus.) Together with Theorem 1.5 of[1], this proves the existence of counterexamples to Morimoto's Conjecture(Conjecture 1.5 of [2]). [1] Tsuyoshi Kobayashi and Yo'av Rieck. On the growthrate of the tunnel number of knots. J. Reine Angew. Math., 592:63--78, 2006.[2] Kanji Morimoto. On the super additivity of tunnel number of knots.Math.Ann., 317(3):489--508, 2000.

Knot exteriors with additive Heegaard genus and Morimoto’s Conjecture