Constructing knot tunnels using giant steps

TL;DR

This paper applies a new theoretical framework to analyze the construction of knot tunnels via giant steps, revealing the uniqueness or multiplicity of minimal construction sequences for different tunnels.

Contribution

It extends previous work by using the tree of knot tunnels to quantify the number of minimal sequences of tunnel moves for constructing knot tunnels.

Findings

For some tunnels, the minimal sequence of moves is unique.

Most tunnels have multiple minimal construction sequences.

The theory provides a method to count these sequences.

Abstract

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.