\section{There's more!} \label{sec:more}
There's more! In fact, OU tangles and OU ideas seem prevalent in knot
theory, even though it seems that nobody collected all these ideas
together before. In fact, possibly the most
important contribution of this paper is the observation that, in addition to the detailed examples that we studied throughout, everything mentioned in
this section is OU-related.
\subsection{Weakening the Bond}
The Gliding Fheorem (\ref{fhm:every}) fails because the bond between the strands of
a single crossing is too strong; they cannot be separated to be taken
for rides along other strands in an independent manner: when the U and the
O of a UO interval belong to the same crossing, one cannot glide them
independently of each other and across each other as the glide move of
Figure~\ref{fig:Gliding} dictates. So we seek to weaken this bond.
\Needspace{39mm} % 38mm is not enough.
\parpic[r]{\input{figs/R.pdf_t}}
One way to do so is with algebra. One aims to construct invariants of
tangles by placing ``R-matrices'' on positive crossings (and their
inverses on negative crossings). An R-matrix is an element
$R=\sum_i{b_i\otimes a_i}\in H\otimes H$ in the tensor square of some
algebra $H$, and its $b_i$ side is placed on the O side of the crossing
while its $a_i$ side is put on the U side. This done, one multiplies the
algebra elements seen on each strand in the order in which they appear
along it, and the hope is that the result would be an invariant of the
tangle, living in $H^{\otimes S}$ where $S$ is the set of strands.
In this context ``O'' becomes ``$b_i$'' and ``U'' becomes ``$a_i$'',
and the bond between O and U is nearly severed --- within a long
product, given the appropriate commutation relations, $b_i$'s can
be commuted against $a_i$'s whether or not they originally came from
the same crossing. Further effort is needed in order to make use of
this fact, and it is beyond the scope of this summary to reproduce
this effort here. Yet the result becomes ``something from nothing'':
given relatively little input, a construction of an R-matrix and the
algebra $H$ in which it lives. This construction is better known as
``the Drinfel'd double construction''. See more at~\cite{Talk:OU} and
hopefully in a future publication.
\Needspace{40mm} % 39mm is not enough.
\parpic[r]{\input{figs/Surgery.pdf_t}}
Another way to weaken the bond between the O side and the U side of a single crossing is to represent crossings
using surgery. A quick summary is on the right: a crossing can be created using a $+1$ surgery on a loop
surrounding the two strands to be crossed, and that loop is relatively loose bond between these two strands,
for in itself it can be pushed around.
This story is imprecise and incomplete: Imprecise because strictly speaking, the surgery shown created two
crossings and not just one. Incomplete in several ways; the most important is that general surgeries can
change the ambient space from $S^3$ into another 3-manifold, and thus to properly pursue this idea one must
study an appropriate class of tangles in manifolds. See more at~\cite{ThurstonD:Sutured} and
hopefully in a future publication.
\subsection{Prior Art}
Milnor attempted to classify links up to link homotopy by introducing the
``reduced peripheral system''~\cite{Milnor:LinkGroups}. Unfortunately,
this is only a complete invariant of homotopy links with up to three
components. In~\cite{AudouxMeilhan:PeripheralSystems} Audoux and Meilhan
use a gliding-type algorithm and OU links to give a full analysis of the
kernel of this invariant: namely, they prove that the reduced peripheral
system is a complete invariant of {\em welded} homotopy links (welded
links up to {\em self-virtualization} equivalence). Welded links are
closely related to knotted ribbon tori in $\bbR^4$, and homotopy links
map non-injectively into welded homotopy links by ``spinning around a
plane.'' See~\cite[Definition~1.7]{AudouxMeilhan:PeripheralSystems},
where OU links are called {\em sorted}. See
also~\cite[Definition~4.15]{AudoxBellingeriMeilhanWagner:WeldedStringLinks}
where they are called ``ascending''.
An earlier occurrence of OU ideas in the context of w-tangles is in the
paper~\cite{KBH} whose theme is the separation of hoops, that can only
go Under, from balloons, that go both Under and Over (so~\cite{KBH}
is a bit less ``pure'', as the balloons are not quite O). Later within
the same paper, and also within \cite{WKO2, WKO3, WKO4}, the associated
graded space of the space of w-tangles is studied, the space $\calA^w$
of ``arrow diagrams modulo the TC relation''. Furthermore that space is
studied using various ``Heads then Tails'' techniques, which in the language of the
current paper, correspond to UO presentations (not OU, but of course,
it's essentially the same). See especially~\cite[Section~2.4]{WKO4}.
\Needspace{15mm} % 14mm is not enough.
\parpic[r]{\input{figs/TH.pdf_t}}
An even earlier occurrence of OU ideas, in the associated
graded $\calA^v$ context for virtual tangles, occurs in a
very well-hidden way within Enriquez' work on quantization
of Lie bialgebras~\cite{Enriquez:UniversalAlgebras,
Enriquez:QuantizationFunctors}. For example, his ``universal
algebras''~\cite[Section~1.3.2]{Enriquez:QuantizationFunctors} are
isomorphic to the space $\calA^v_{OU}$ of arrow diagrams as on the right,
in which all arrow tails occur before all arrow heads (that's OU!), and
is endowed with the product that $\calA^v_{OU}$ inherits from the stacking
product of $\calA^v$ (which is the analogue of the product used in our
paper). We are afraid that there aren't excellent introductions available
on $\calA^v$ and its relationship with virtual tangles. Hopefully we
will write one one day. Until then, some information is in~\cite{WKO2}
and in lecture series such as~\cite{Caen, MasterClass}. We also hope to
one day explain the Enriquez work as the construction of a ``homomorphic
expansion''~\cite{WKO1} for the space of virtual OU / acyclic tangles.
If $\frakg=\fraka^\star\bowtie\fraka$ is the double of a Lie bialgebra
$\fraka$, there is a standard interpretation of $\calA^v$ as a space
of formulas for elements in tensor powers $\calU(\frakg)^{\otimes n}$
of the universal enveloping algebra $\calU(\frakg)$ of $\frakg$. Within
this context, arrow tails (or ``O'') correspond to $\fraka^\star$
and arrow heads (or ``U'') correspond to $\fraka$, and the O then U
theme of this paper corresponds to the ``polarization'' isomorphism
$\calU(\frakg)\cong\calU(\fraka^\star)\otimes\calU(\fraka)$, which is
a consequence of the PBW theorem. In itself, the polarization isomorphism is central to all approaches to the
quantization of Lie bialgebras~\cite{EtingofKazhdan:BialgebrasI, Severa:BialgebrasRevisited}.