The over then under (OU) tangle diagrams provide a new framework for knot theory.
To illustrate the power of this way of thinking we show that every braid
has a unique minimal OU diagram. As a corollary we produce a new type of graphs that that are canonically associated to braids.
These results are also generalized to virtual braids.
Many techniques in knot theory can be understood in terms of OU diagrams, even though
for knots such diagrams may not exist in the literal sense. We argue that these ideas shed new light upon
subjects such as the Drinfel'd double construction of quantum group theory and the quantization of Lie bialgebras.
%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.