Summary: UNKNOTTING TUNNELS AND SEIFERT SURFACES
MARTIN SCHARLEMANN AND ABIGAIL THOMPSON
ABSTRACT. Let K be a knot with an unknotting tunnel and suppose
that K is not a 2-bridge knot. There is an invariant p q ¾ É 2 , p
odd, defined for the pair ´K µ.
The invariant has interestinggeometric properties: It is often straight-
forward to calculate; e. g. for K a torus knot and an annulus-spanning
arc, ´K µ 1. Although is defined abstractly, it is naturally revealed
when K is put in thin position. If 1 then there is a minimal
genus Seifert surface F for K such that the tunnel can be slid and iso-
toped to lie on F. One consequence: if ´K µ 1 then genus´Kµ 1.
This confirms a conjecture of Goda and Teragaito for pairs ´K µ with
´K µ 1.
1. INTRODUCTORY COMMENTS
In [GST] the following conjecture of Morimoto's was established: if a
knot K S3 has a single unknotting tunnel , then 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 one can simplify its presentation until the tunnel is either
a level arc or a level circuit.