Constructing knot tunnels using giant steps

Abstract

In 2000, Goda, Scharlemann, and Thompson described a general construction of all tunnels of number 1 knots using tunnel The theory of tunnels introduced by Cho and McCullough provides a combinatorial approach to understanding moves. We use it to calculate the number of distinct minimal sequences of such 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 will have many such constructions. Finally, we give a characterization of the tunnels with a unique minimal sequence.