Middle tunnels by splitting

TL;DR

This paper studies middle tunnels of genus-1 1-bridge knots in the 3-sphere, generalizing previous constructions and calculating slope invariants, including the sequence for a known example.

Contribution

It extends the construction of middle tunnels for (1,1)-knots and computes their slope invariants, enriching the understanding of knot tunnel structures.

Findings

Most torus knots have a middle tunnel.

Generalized construction for middle tunnels of (1,1)-knots.

Calculated slope sequences for these tunnels.

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. We generalize their construction and calculate the slope invariants for the resulting middle tunnels. In particular, we obtain the slope sequence of the original example of Goda, Hayashi, and Ishihara.