On the growth rate of tunnel number of knots

Abstract

Given a knot $K$ in a closed orientable manifold $M$ we define the growth rate of the tunnel number of $K$ to be $g{r}_t(K)={\mathrm{limsup}}_{n\to \infty }\frac{t(nK)-nt(K)}{n-1}$. 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 $S^3$ is never asymptotically super additive. The main result gives conditions that imply falsehood of Morimoto's Conjecture.

19 pages, 8 figures

Cited by in corpus (1)

• Heegaard genus of the connected sum of m-small knots