We prove that if an alternating knot has unknotting number one, then there
exists an unknotting crossing in any alternating diagram. This is done by
showing that the obstruction to unknotting number one developed by Greene in
his work on alternating 3-braid knots is sufficient to identify all unknotting
number one alternating knots. As a consequence, we also get a converse to the
Montesinos trick: an alternating knot has unknotting number one if its branched
double cover arises as half-integer surgery on a knot in S3. We also reprove
a characterisation of almost-alternating diagrams of the unknot originally due
to Tsukamoto.Comment: 38 pages, 15 figures. This a significant revision of the first
version. A proof of Tsukamoto's work on almost-alternating diagrams of the
unknot is now included. There is also an additional formulation of the main
theorem which makes precise the signs of the unknotting crossings and the
resulting half-integer surgeries. Some comments on potential further
questions have also been adde