Unknotting tunnels and Seifert surfaces

Abstract

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho =p/q\in \mathbb{Q}/2\mathbb{Z}$, $p$ odd, defined for the pair $(K,\gamma )$. The invariant $\rho$ has interesting geometric properties: It is often straightforward to calculate; e. g. 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 $\rho \ne 1$ 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: if $\rho (K,\gamma )\ne 1$ then $genus(K)>1$. This confirms a conjecture of Goda and Teragaito for pairs $(K,\gamma )$ with $\rho (K,\gamma )\ne 1$.

29 pages, 20 figures