Constructing knot tunnels using giant steps
  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. 
Sangbum Cho, Darryl McCullough