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, p odd, defined for the pair (K, γ). The invariant ρ has interesting geometric properties: It is often straightforward to calculate; e. g. 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 6= 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: if ρ(K, γ) 6= 1 then genus(K) > 1. This confirms a conjecture of Goda and Teragaito for pairs (K, γ) with ρ(K, γ) 6= 1. 1. Introductory comments In [GST] the following conjecture of Morimoto’s was established: if a knot K ⊂ S has a single unknotting tunnel γ, then γ can be moved to be level with respect to the natural height function on K given by a minimal bridge presentation of K. The repeated theme of the proof is that by “thinning” the 1-complex K ∪ γ one can simplify its presentation until the tunnel is either a level arc or a level circuit. The present paper was originally motivated by two questions. One was a rather specialized conjecture of Goda and Teragaito: must a hyperbolic knot which has both genus and tunnel number one necessarily be a 2-bridge knot? A second question was this: Once the thinning process used in the proof of [GST] stops because the tunnel becomes level, can thin position arguments still tell us more? With respect to the second question, it turns out that there is an obstruction to further useful motion of γ that can be expressed as an element ρ ∈ Q 2Z . Surprisingly, further investigation showed that, so long as K is not 2-bridge, the obstruction ρ can be defined in a way completely independent of thin position and thereby can be viewed as an invariant of the pair (K, γ). Moreover, this apparently new invariant has useful properties: It is not hard to calculate. If ρ 6= 1, then the Date: February 1, 2008. Research supported in part by grants from the National Science Foundation.