Tunnel Number One Knots Satisfy the Poenaru Conjecture

The tunnel number of a PL knot K is the minimum number of PL one-cells which must be attached in order that the regular neighborhood of the resulting complex has complement a handlebody [2]. It is easy to see that n-bridge knots have tunnel numbers (n 1) and torus knots have tunnel number one. More difficult is finding knots which have higher tunnel number. If a knot has tunnel number one, it will be a one-relator knot and therefore prime [6] (for a geometric proof that tunnel number one knots are prime see 2.2). Among prime knots, those with nonvanishing second elementary ideal are not one-relator and therefore not tunnel number one. It seems a good but difficult conjecture that one-relator knots coincide with tunnel number one knots. The goal here is to show that tunnel number one knots satisfy two other properties: they are doubly prime (that is, they cannot be written as the join of two prime tangles) and they satisfy the Poenaru conjecture (that is, no 2k + I longitudes of the knot bound an incompressible, boundary incompressible planar surface in the complement of a tubular neighborhood of the knot). The first property is really a curiosity-it is easily proven using the techniques developed elsewhere in the paper and so is included here. The second propeny is the crucial difficult step in the solution of the genus two Schoenflies conjecture [8].