\section{Introduction}

Brilliant wrong ideas should not be buried or forgotten. Instead, they should be mined for the gold that lies underneath the layer of wrong. 

In this paper we introduce {\em Over then Under} (OU) tangles, a class of oriented tangles in which each strand travels through all of its under-crossings before any of its over-crossings: see Figure~\ref{fig:OUExamples} for some examples and Definition~\ref{def:OU} for details. This is equivalent to the notion of {\em ascending} tangles in~\cite[Definition~4.15]{AudoxBellingeriMeilhanWagner:WeldedStringLinks}, also called {\em sorted} in \cite[Definition~1.7]{AudouxMeilhan:PeripheralSystems} in the context of welded homotopy links.

The key, but incorrect, observation at the core of this paper -- explained in Section~\ref{sec:Gliding} -- is that every tangle can be brought to OU-form using a sequence of {\em glide moves}: specific isotopies designed to eliminate any ``forbidden sequences'' of crossings along a strand. The argument is compelling, and has sweeping consequences, including the -- clearly false -- corollary that every knot is trivial. On closer look, one notices that in certain special cases of a strand crossing itself, the glide moves fail. 

There is, however, much to salvage from the failure of the gliding idea: the argument of Section~\ref{sec:Gliding} holds for braids, and every braid -- when considered as a tangle -- has a unique OU form. Hence, the OU form is a {\bf separating braid invariant}. We also prove that in fact, tangles which can be brought to OU form are precisely braids, using the identification of the braid group with the mapping class group of a punctured disc (see eg \cite[Theorem 1]{BirmanBrendle:BraidsSurvey}, also explained in Section~\ref{sec:classical}.) 

Even better, the gliding argument extends to virtual braids to show that every virtual braid has a unique OU form when it is regarded as a virtual tangle. With extra work we find that this OU form is a {\bf complete invariant for virtual braids}. This is the subject of Section~\ref{sec:virtual}.

In Section~\ref{sec:comp} we present Mathematica implementations, including tabulations of virtual pure braids and classical braids.

In Section~\ref{sec:more} we review a range of other instances in the literature where ``OU ideas'' play a role: Drinfeld's double construction in quantum groups, a classification of welded homotopy links by Adoux and Meilhan  \cite{AudouxMeilhan:PeripheralSystems}, Enriquez's work on the quantization of Lie bialgebras \cite{Enriquez:UniversalAlgebras,
Enriquez:QuantizationFunctors}, and earlier work of the authors.

All tangle diagrams in this paper are {\em open and oriented}:
Their components are always oriented intervals and never circles. For simplicity and definiteness, all
tangles in this paper are unframed: we allow all Reidemeister 1 (R1) moves, though this is not strictly
necessary and similar results also hold in the framed case.


\section{OU Tangles and Gliding}\label{sec:Gliding}

\begin{definition} \label{def:OU} An Over-then-Under
(OU) tangle diagram
is a tangle whose strands complete all of their over crossings before any
of their under crossings, and an OU tangle is an oriented tangle that can
be represented by an OU tangle diagram. 

In detail, an OU tangle diagram
is an oriented tangle diagram each of whose strands can be divided in
two by a ``transition point'', sometimes indicated with a bow tie symbol
$\bowtie$, such that in the first part (before the transition) it is the
``over'' strand in every crossing it goes through, and in the second part
(after the transition) it is the ``under'' strand in every crossing it
goes through, so a journey through each strand looks like an OO\ldots
O($\bowtie$)UU\ldots U sequence of crossings. Some examples are shown in
Figure~\ref{fig:OUExamples}.
\end{definition}

\begin{remark} Loosely, an
OU tangle is the ``opposite'' of an alternating tangle: crossings along each strand read OOOUUU rather than OUOUOU.
\end{remark}

\begin{figure}%[t]
\[ \input{figs/OUExamples.pdf_t}\includegraphics[height=0.875in]{BorrBraidOUDiagram2.pdf} \]
\caption{The tangle diagram (A) is OU as strand 1 is all ``over''
  (so it has an empty ``U'' part) and strand 2 is all ``under'' (so it
  has an empty ``O'' part). The tangle diagram (B) is not
  OU: strand 1 is O then U, but strand 2 is U then O. Yet the tangle
  represented by (B) is OU because it is also represented
  by (C), which is OU. The diagram (D) is again OU; which familiar tangle does it represent?
} \label{fig:OUExamples}
\end{figure}

The following Fheorem (false theorem), while unfortunately not true, illustrates the idea and potential of {\em gliding}:

\begin{fheorem}[Gliding] \label{fhm:every} Every tangle is an OU tangle.
\end{fheorem}

%\parpic[r]{\parbox{3in}{
%  \resizebox{\linewidth}{!}{\input{figs/froof.pdf_t}}
%  \caption{Glide moves.} \label{fig:Gliding}
%}}
\par\noindent{\it Froof.} As in Figure~\ref{fig:Gliding}, the froof
is frivial. Assume first that  strands 1 and 2 are already in OU form
(meaning, all their O crossings come before all their U ones) but strand 3
still needs fixing, because at some point it goes through two crossings, first under and then over,
as on the left of Figure~\ref{fig:Gliding}. Simply glide strand 1 forward along and over 3 and
glide strand 2 back and under 3 as in Figure~\ref{fig:Gliding}, and the UO interval
along 3 is fixed, and nothing is broken on strands 1 and 2 --- strand
1 was over and remains over (more precisely, the part of strand 1 that
is shown here is the ``O'' part), and strand 2 is under and remains
under.

In fact, it doesn't matter if strands 1 and 2 are already in OU form because as shown in the
second part of Figure~\ref{fig:Gliding}, glide moves can be performed ``in bulk''. All that the
fixing of strand 3 does to strands 1 and 2 is to replace an O by an OOO on strand 1 and a U by a
UUU on strand 2, and this does not increase their complexity as UU\ldots UOO\ldots O sequences can
be fixed in one go using bulk glide moves.\fed

\begin{figure}
\[ \input{figs/froof.pdf_t} \]
\caption{Glide moves between two crossings and bulk glide moves.} \label{fig:Gliding}
\end{figure}

\begin{forollary}\label{for:KnotsTrivial}
All long knots are trivial.
\end{forollary}

\par\noindent{\it Froof.}
It is clear that any OU tangle on a single strand is trivial.
\fed


\Needspace{19mm} % 18mm is not enough.
\parpic[r]{\input{figs/UO.pdf_t}}
\begin{discussion} \label{disc:froofs1}
Forollary~\ref{for:KnotsTrivial} is clearly false. For
the Gliding Fheorem (\ref{fhm:every}) is a Theorem
with its T replaced with an F and its froof is a spoof with a leaky Halmos. Indeed, while
everything we said about glide moves holds true, there is another way a strand may be U and
then O: the U and O may be parts of a single crossing, as on the right, instead of belonging to two distinct
crossings, as in the left hand side of the glide move.
\end{discussion}

\parpic[r]{\input{figs/UUOO.pdf_t}} It is tempting to dismiss this with
``it's only a Reidemeister 1 (R1) issue, so one may glide all
kinks to the tail of a strand and count them at the end''.  Except the
same issue can arise in ``bulk'' UU\ldots UOO\ldots O situations (as now
on the right), where it cannot be easily dismissed.  One may attempt
to resolve the UUOO situation on the right using single (non-bulk)
glide moves. We have no theoretical reason to expect this to work as the
lengths of UU\ldots U and OO\ldots O sequences may build up faster than
they are sorted. And indeed, it doesn't work. Figure~\ref{fig:swirls}
shows what happens.



\begin{figure}
\def\rO{{\red O}}\def\rU{{\red U}}\def\b{{\color{black}$\bullet$}}
\resizebox{\linewidth}{!}{\input{figs/swirls.pdf_t}}
\caption{
  An attempt to fix a non-OU tangle diagram. In each step we use a
  single glide move to fix the first UO sequence encountered on strand
  1 (we mark it with a $\bullet$), but things get progressively more
  complicated. The O/U sequences below the diagrams are listed from the
  perspective of strand 1.
} \label{fig:swirls}
\end{figure}


It is true (and also follows from Corollary~\ref{cor:cOUisBraids})
that the only 1-component OU tangle is the trivial one.
\endpar{\ref{disc:froofs1}}

\begin{discussion} \label{disc:Options}
What can we salvage from the disappointing failure of gliding?
There are many options to consider. Perhaps Fheorem~\ref{fhm:every} becomes true if
we restrict to some subset of the set of all tangles? (Braids,
Section~\ref{sec:classical}). Or perhaps if we extend to some
superset? Or in a subset of a superset? (Virtual
braids, Section~\ref{sec:virtual}). Perhaps we ought to look at some
form of finite-type completion? Perhaps we should look at tangles
in manifolds? At quotients of the space of tangles? At some
combinations of these?

In the authors' opinion it is worthwhile to explore
these options. In fact, 
many of these options have already been explored, each in a different
context and without the realization that these different contexts share
a common theme: see Section~\ref{sec:more}. \endpar{\ref{disc:Options}}
\end{discussion}