Cabling sequences of tunnels of torus knots

TL;DR

This paper calculates the unique parameter sequences that describe all tunnels of torus knots, refining the theory of knot tunnels and providing a detailed classification within this family.

Contribution

It extends the parameterization of knot tunnels to all torus knot tunnels, offering explicit invariants for this class.

Findings

Complete parameter sequences for all torus knot tunnels

Refinement of the tunnel classification theory

Explicit construction procedures for torus knot tunnels

Abstract

This is the second of three papers that refine and extend portions of our earlier preprint, "The depth of a knot tunnel." Together, they rework the entire preprint. The theory of tunnel number 1 knots that we introduced in "The tree of knot tunnels" yields a parameterization in which each tunnel is described uniquely by a finite sequence of rational parameters and a finite sequence of 0's and 1's, that together encode a procedure for constructing the knot and tunnel. In this paper we calculate these invariants for all tunnels of torus knots.