On the growth rate of tunnel number of knots

Details On the growth rate of tunnel number of knots On the growth rate of tunnel number of knots

2004-02-03

arxiv.org/abs/math/0402025v1

Mathematics

Geometric Topology

19 pages, 8 figures

Scientific paper

Given a knot $K$ in a closed orientable manifold $M$ we define the growth rate of the tunnel number of $K$ to be $gr_t(K) = \limsup_{n \to \infty} \frac{t(nK) - n t(K)}{n-1}$. 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.

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