  A {\em good drawing\/} of $K_n$ is a drawing of the complete graph with $n$
vertices in the sphere such that: no two edges with a common end cross; no two
edges cross more than once; and no three edges all cross at the same point.
Gioan's Theorem asserts that any two good drawings of $K_n$ that have the same
rotations of incident edges at every vertex are equivalent up to Reidemeister
moves. At the time of preparation, 10 years had passed between the statement in
the WG 2005 conference proceedings and our interest in the proposition. Shortly
after we completed our preprint, Gioan independently completed a preprint.
