Knot exteriors with additive Heegaard genus and Morimoto's Conjecture
Abstract
Given integers gi > 1 (i=1,...,n) we prove that there exist infinitely may knots Ki in S3 so that g(E(Ki)) = gi and the Heegaard genus of the exterior of the connected sum of K1,...,Kn is the sum the Heegaard genera of K1,...,Kn, that is: g(E(K1#...#Kn)) = g(E(K1)) +...+ g(E(Kn)). (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 growth rate 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.
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