  This is the first of three papers that refine and extend portions of our
earlier preprint, "Depth of a knot tunnel." Together, they rework the entire
preprint.
  H. Goda, M. Scharlemann, and A. Thompson described a general construction of
all tunnels of all tunnel number 1 knots using "tunnel moves". We apply the
theory that we introduced in "The tree of knot tunnels" to study this
construction. In particular, we use it to calculate the number of distinct
minimal sequences of tunnel moves that can produce a given tunnel. As a
consequence, we see that for a sparse infinite set of tunnels, the minimal
sequence is unique, but generically a tunnel will have many such constructions.
