On the Growth Rate of Tunnel Number of Knots

Given a knot K in a closed orientable manifold M we define the growth rate of the tunnel number of K to be grt(K) = lim supn→∞ t(nK)−nt(K) n−1 . As our main result we prove that the Heegaard genus of M is strictly less than the Heegaard genus of the knot exterior if and only if the growth rate is less than 1. In particular this shows that a non-trivial knot in S is never asymptotically super additive. The main result gives conditions that imply falsehood of Morimoto’s Conjecture.