\section{Fheorems and Froofs}

\begin{discussion} \label{disc:first}
In Definition~\ref{def:OU} we define ``Over then Under" (OU) tangles. Then in
Fheorem~\ref{fhm:every} we frove\footnote{Please bear with our spelling.}
that every tangle is an OU tangle and in Theorem~\ref{thm:unique} we
argue that if a tangle is OU, its OU diagram is essentially unique. Hence
(Forollary~\ref{for:classification}) the classification of knots is
completely frivial: a knot is a long knot which is a tangle with one
component which can be brought to a unique OU form!

This discussion continues as Discussion~\ref{disc:froofs1}.\endpar{\ref{disc:first}}
\end{discussion}

\begin{convention} All tangle diagrams in this paper are ``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.
\end{convention}

%An Over-then-Under (OU) tangle
%diagram\footnote{Equivalent to ``ascending''
%in~\cite[Definition~4.15]{AudoxBellingeriMeilhanWagner:WeldedStringLinks}.}
%is an oriented tangle diagram that contains no Under-then-Over
%(UO) intervals --- subintervals of strands that begin at a crossing
%as the under strand (U), and end at a crossing as the over strand (O).

\begin{definition} \label{def:OU} An Over-then-Under
(OU) tangle diagram\footnote{Equivalent to ``ascending''
in~\cite[Definition~4.15]{AudoxBellingeriMeilhanWagner:WeldedStringLinks}.}
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 in
Figure~\ref{fig:OUExamples}.
\end{definition}

\begin{remark} Loosely, an
OU tangle is the ``opposite'' of an alternating tangle: it is 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}

\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 completely 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}

Note that if a tangle diagram is OU then no Reidemeister 3 (R3) moves can be performed on it without breaking the
OU property --- if
one side of an R3 move is OU, the other necessarily isn't. This
suggests that perhaps an OU form of a tangle diagram is unique up to Reidemeister 2 (R2) moves. We
aim to prove this next.

\begin{theorem} \label{thm:unique} When glide moves (Figure~\ref{fig:Gliding}) are used to fix a
tangle diagram to be OU, the result is independent of the order in which they are performed.
\end{theorem}

\par\noindent{\it Froof.} When UO intervals are apart from each
other, their fixing is clearly independent. It remains to see what
happens when UO intervals are adjacent, and there are only two
distinct cases to consider. Both of these cases are shown
in Figure~\ref{fig:unique} along with their OU fixes, which are clearly
independent of the order in which the glide moves are performed.  \fed

\begin{figure}
\[ \resizebox{0.92\linewidth}{!}{\input{figs/unique.pdf_t}} \]
\caption{Two possibilities for ``interacting'' UO intervals (each marked with a~$\bullet$ symbol).}
\label{fig:unique}
\end{figure}

\begin{definition} A tangle diagram is called reduced if its crossing number cannot be reduced
using only R1 and R2 moves.
\end{definition}

\begin{forollary}[Separation of Tangles] \label{for:UniqueReduced}
Every tangle has a unique reduced OU diagram.
\end{forollary}

\Needspace{34mm}
\parpic[r]{\input{figs/R3.pdf_t}}
\par\noindent{\it Froof.} If two tangle diagrams differ by an R3 move then
exactly one of them has a UO interval within the scope of the R3 move, and
its elimination via a glide move (which may as well be performed first)
yields the other diagram, up to an R2 move (picture on right). It is
also easy to see that R1 and/or R2 moves before a glide become R1 and/or
R2 moves after the glide (or they make the glide move redundant, see
examples in Figure~\ref{fig:R1R2}), so the end result of the gliding
process of a tangle is unique modulo R1 and R2 moves. Finally it is easy
to check that within any equivalence class of tangle diagrams modulo R1
and R2 moves there is a unique reduced representative. \fed

\begin{figure}
\[ \input{figs/R1R2.pdf_t} \]
\caption{R1 and R2 moves ``commute'' with glides (A), or they
make glides redundant (B), (C).} \label{fig:R1R2}
\end{figure}

\begin{forollary}[of Forollary~\ref{for:UniqueReduced}] \label{for:classification}
Every long knot has a unique reduced OU diagram, and 
therefore, the classification of knots is frivial.
\end{forollary}

\par\noindent{\it Froof.} A long knot is the same as a 1-strand tangle.\fed

\Needspace{19mm} % 18mm is not enough.
\parpic[r]{\input{figs/UO.pdf_t}}
\begin{discussion} \label{disc:froofs1}
Forollary~\ref{for:classification} seems too good to be
true, and in fact, it isn't. 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. For indeed, while
everything we said about glide moves holds true, there is another way a strand may be U and
then O --- those 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, and one may anyway 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.

In fact, it is easy to see (and follows from Corollary~\ref{cor:cOUisBraids})
that the only 1-component OU tangle is the trivial, making the failure
of Forollary~\ref{for:classification} quite spectacular.
\endpar{\ref{disc:froofs1}}

\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}

\Needspace{17mm} % 16mm is not enough.
\parpic[r]{\input{figs/OmittedCase.pdf_t}}
\begin{discussion} Similarly, the froof of Theorem~\ref{thm:unique} is
faulty because one case of adjacent UO intervals (shown on the right)
is omitted. Yet in fact, Theorem~\ref{thm:unique} holds true as stated,
for if the piece on the right occurs within a tangle diagram, it cannot be
fixed to be OU using glide moves. This along with Theorem~\ref{thm:unique}
itself follow from Theorem~\ref{thm:acyclic}, proven in the next section.
\end{discussion}

\begin{discussion} The Separation of Tangles Forollary
(\ref{for:UniqueReduced}) is of course false as stated, but only
because it relies on the Gliding Fheorem (\ref{fhm:every}), and in
fact there is some truth in its froof. See Corollaries~\ref{cor:braids}
and~\ref{cor:vbraids}.
\end{discussion}

\begin{discussion} \label{disc:Options}
While the fheorems of this section are false, there is a bit of truth
in each of their froofs and it would be a shame to lose that bit for
the bit of falsehood in them. Can we still save something from this situation?

There are many options to consider. Perhaps things become true if
we restrict to some subsets of the set of all tangles? (Braids,
Section~\ref{ssec:classical}). Perhaps if we extend to some
superset? Perhaps if we consider some subset of a superset? (Virtual
braids, Section~\ref{ssec:virtual}). Perhaps we ought to look at some
form of a finite-type completion? Perhaps we should look at tangles
in manifolds? At quotients of the space of tangles? At images? At some
combinations of these options?

The overall theme of this paper is that 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}