Unknotting genus one knots
Alexander Coward
Sydney University
Abstract
An unknotting crossing change on a knot is a way of passing the knot through itself so that the resulting curve is unknotted. In general it is very hard to determine whether a given knot admits an unknotting crossing change. In this talk we address a related question: Given a knot which admits an unknotting crossing change, how many different ways are there of unknotting it? We prove that for all genus one knots an unknotting crossing change is unique, up to a suitable notion of equivalence, with the exception of the figure-eight knot which admits exactly two unknotting crossing changes. The proof will be presented so as to be understandable to non-experts. This is joint work with Marc Lackenby.