Identifying tunnel number one knots

Abstract

Let $K$ be a knot in $S^{3}$ . The tunnel number $t(K)$ of $K$ is the minimal number of mutually disjoint arcs $\{\tau_{i}\}$ "properly embedded" in the pair $(S^{3}, K)$ such that the complement of an open regular neighbourhood of $K\cup(\cup\tau_{\mathfrak{i}})$ is a handlebody. In the above, if the arc system consists of only one arc, it is called an unknotting tunnel for K. $K$ is said to have a $(g, b)$ -decomPosition if there is a genus $g$ Heegaard splitting $\{W_{1}, W_{2}\}$ of $S^{3}$ such that $K$ intersects $W_{i}(i=1,2)$ in a b-string trivial arc system (cf. $[D$, MS]). If a knot $K$ has a $(g, b)$ -decomposition, then t(K)$g+b--1. In particular; if $K$ admits a $(1, 1)$ -decomposition then it has tunnel number one; however, it is shown by [MR, MSY, Yol] that the converse does not hold.