  For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle
tunnel is a tunnel that is not an upper or lower tunnel for some
(1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot
examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we
generalized their construction and calculated the slope invariants for the
resulting examples. We give an iterated version of the construction that
produces many more examples, and calculate their slope invariants. If one
starts with the trivial knot, the iterated constructions produce all the
2-bridge knots, giving a new calculation of the slope invariants of their
tunnels. In the final section we compile a list of the known possibilities for
the set of tunnels of a given tunnel number 1 knot.
