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.
Cite this article
Alexander Coward, Marc Lackenby, Unknotting genus one knots. Comment. Math. Helv. 86 (2011), no. 2, pp. 383–399DOI 10.4171/CMH/227