Knots with and and Morimotoʹs Conjecture

We show that there exist knots K ⊂ S 3 with g ( E ( K ) ) = 2 and g ( E ( K # K # K ) ) = 6 . Together with [Tsuyoshi Kobayashi, Yoʹav Rieck, On the growth rate of the tunnel number of knots, J. Reine Angew. Math. 592 (2006) 63–78, Theorem 1.5], this proves existence of counterexamples to Morimotoʹs Conjecture [Kanji Morimoto, On the super additivity of tunnel number of knots, Math. Ann. 317 (3) (2000) 489–508]. This is a special case of [Tsuyoshi Kobayashi, Yoʹav Rieck, Knot exteriors with additive Heegaard genus and Morimotoʹs Conjecture, Algebr. Geom. Topol. 8 (2008) 953–969, preprint version available at http://arxiv.org/abs/math.GT/0701765, 2007].