Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a $2$-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, with $p$ odd, defined for the pair $(K, \gamma)$.
The invariant $\rho$ has interesting geometric properties. It is often straightforward to calculate; for example, for $K$ a torus knot and $\gamma$ an annulus-spanning arc, $\rho(K, \gamma) = 1$. Although $\rho$ is defined abstractly, it is naturally revealed when $K \cup \gamma$ is put in thin position. If $\rho \neq 1$ then there is a minimal-genus Seifert surface $F$ for $K$ such that the tunnel $\gamma$ can be slid and isotoped to lie on $F$. One consequence is that if $\rho(K, \gamma) \neq 1$ then $\mathrm{genus}(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \gamma)$ with $\rho(K, \gamma) \neq 1$.
