Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel γ and suppose that K is not a 2‐bridge knot. There is an invariant ρ = p / q ∈ Q/2Z, with p odd, defined for the pair ( K , γ). The invariant ρ has interesting geometric properties. It is often straightforward to calculate; for example, for K a torus knot and γ an annulus‐spanning arc, ρ( K , γ) = 1. Although ρ is defined abstractly, it is naturally revealed when K ∪ γ is put in thin position. If ρ ≠ 1 then there is a minimal‐genus Seifert surface F for K such that the tunnel γ can be slid and isotoped to lie on F . One consequence is that if ρ( K , γ) ≠ 1 then K > 1. This confirms a conjecture of Goda and Teragaito for pairs ( K , γ) with ρ( K , γ) ≠ 1. 2000 Mathematics Subject Classification 57M25, 57M27.