On the growth rate of tunnel number of knots

TL;DR

This paper investigates the growth rate of the tunnel number of knots in 3-manifolds, establishing a key relationship between this growth rate and the Heegaard genus, with implications for Morimoto's Conjecture.

Contribution

It proves that the Heegaard genus relation characterizes the growth rate of the tunnel number, providing new insights into knot complexity and disproving asymptotic super additivity for non-trivial knots in S^3.

Findings

Heegaard genus of the manifold is less than that of the knot exterior iff the growth rate is less than 1.

Non-trivial knots in S^3 are never asymptotically super additive.

Conditions are provided that challenge Morimoto's Conjecture.

Abstract

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.