\section{Introduction}

In this paper we study {\em Over then Under} (OU) tangles,
a class of oriented tangles in which each strand travels through
all of its over-crossings before any of its under-crossings. See
Figure~\ref{fig:OUExamples} for examples; the full definition is given in Section~\ref{sec:Gliding}.

\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}

We present an algorithm (Section~\ref{sec:Gliding}) which brings non-OU
tangle diagrams to OU form using a sequence of {\em glide moves}:
specific isotopies designed to eliminate any ``forbidden sequences''
of crossings along a strand. At first glance it seems to converge for
any tangle diagram; on closer look, however, one notices that in certain
special cases of a strand crossing itself, the glide moves fail.

The goal of this paper is to investigate and review special cases where the gliding algorithm 
does converge. Indeed, when it does, it can be extremely useful, and in fact gliding ideas have been in use in knot theory
and quantum algebra for some
 time, without being recognised as part of one theme. In our opinion, there is much yet to be gained by
looking for further applications; in this paper, we present two applications in detail.

The gliding algorithm converges 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 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}.

Section~\ref{sec:Assorted} contains some additional comments, mostly on
the relationship between OU tangles and Hopf algebras and on ``Extraction
Graphs'', labeled graphs that are naturally associated with braids
and virtual braids by the process of recovering them from their OU forms.

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 ``gliding ideas'' play a role: the Drinfel'd double
construction in quantum groups, a classification of welded homotopy
links by Audoux 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.

\draftcut \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. 

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.

In greater 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}

Good mathematics is often discovered via wrong proofs and false theorems, which we mine for the truth still contained within. But in academic writing, one presents only the final product. Here, we take a half-page detour from academic tradition to present a Fheorem (false theorem), which, while it ultimately fails, illustrates the idea and potential of {\em gliding}. The reader who prefers tradition should rest assured that everything in the paper from Discussion~\ref{disc:froofs1} onwards is true.

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

\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}

%\Needspace{1mm} % 0mm is not enough.
\parpic[r]{\input{figs/Descending.pdf_t}}
\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 for it must
be descending as in the example on the right.
\fed


\Needspace{20mm} % 19mm is not enough.
\parpic[r]{\input{figs/UO.pdf_t}}
\begin{discussion} \label{disc:froofs1}
Forollary~\ref{for:KnotsTrivial} is clearly false, and so froof of the Gliding Fheorem (\ref{fhm:every}) must be false.
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}