Classification of unknotting tunnels for two bridge knots

TL;DR

This paper classifies all unknotting tunnels for two bridge knots, demonstrating they are isotopic to known examples, thereby completing the classification up to isotopy and homeomorphism.

Contribution

It provides a complete classification of unknotting tunnels for two bridge knots, extending previous results with a comprehensive isotopy and homeomorphism analysis.

Findings

All unknotting tunnels for two bridge knots are isotopic to known ones.

The classification is complete up to isotopies and homeomorphisms.

The result extends Morimoto-Sakuma's previous work.

Abstract

In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to isotopies and homeomorphisms.