Brilliant wrong ideas should not be buried and forgotten. Instead, they
should be mined for the gold that lies underneath the layer of wrong. In
this paper we  explain how ``over then under tangles'' lead to an easy
classification of knots, and under the surface, also to some valid
mathematics: a separation theorem for braids and virtual braids. We end
the paper with an overview of other instances where ``over then under''
ideas play a role: a topological understanding of the Drinfel'd double
construction of quantum group theory, the quantization of Lie bialgebras,
and more.