The spectrum of the growth rate of the tunnel number is infinite

TL;DR

This paper demonstrates that the spectrum of the growth rate of the tunnel number for knots is infinite by constructing hyperbolic knots with growth rates arbitrarily close to 1.

Contribution

It provides the first proof that the spectrum of the growth rate of the tunnel number is infinite by explicit construction.

Findings

Constructed hyperbolic knots with growth rates arbitrarily close to 1

Proved the spectrum of growth rates is infinite

Established the existence of knots with specific asymptotic properties

Abstract

In a previous paper Kobayashi and Rieck defined the growth rate of the tunnel number of a knot $K$, a knot invariant that measures the asymptotic behavior of the tunnel number under iterated connected sum of $K$. We denote the growth rate by $\mbox{gr}_t(K)$. In this paper we construct, for any $\epsilon > 0$, a hyperbolic knots $K \subset S^{3}$ for which $1 - \epsilon < \mbox{gr}_t(K) < 1$. This is the first proof that the spectrum of the growth rate of the tunnel number is infinite.