Iterated splitting and the classification of knot tunnels

TL;DR

This paper introduces an iterative construction method for classifying knot tunnels, computes their slope invariants, and applies the method to generate all 2-bridge knots, enhancing understanding of tunnel structures in knot theory.

Contribution

It presents a new iterative approach to construct and classify knot tunnels, extending previous work and providing comprehensive slope invariant calculations for a wide class of knots.

Findings

Produced many new examples of knot tunnels through iteration.

Calculated slope invariants for these tunnels.

Connected the construction to all 2-bridge knots.

Abstract

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their construction and calculated the slope invariants for the resulting examples. We give an iterated version of the construction that produces many more examples, and calculate their slope invariants. If one starts with the trivial knot, the iterated constructions produce all the 2-bridge knots, giving a new calculation of the slope invariants of their tunnels. In the final section we compile a list of the known possibilities for the set of tunnels of a given tunnel number 1 knot.