Unknotting genus one knots

TL;DR

This paper classifies the unique crossing changes that unknot genus one knots with unknotting number one, revealing a precise count for the figure-eight knot and others, using sutured manifold theory.

Contribution

It provides a complete classification of unknotting crossing changes for genus one knots with unknotting number one, except for the figure-eight knot.

Findings

Exactly one unknotting crossing change for all such knots except the figure-eight

Two unknotting crossing changes for the figure-eight knot

Use of sutured manifold theory and arc complex analysis

Abstract

For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and an analysis of the arc complex of the once-punctured torus.