\draftcut
\section{w-Braids} \label{sec:w-braids}

\begin{quote} \small {\bf Section Summary. }
  \summarybraids
\end{quote}

\subsection{Preliminary: Virtual Braids, or v-Braids.}
\label{subsec:VirtualBraids}
Our main object of study for this section, w-braids, are best
viewed as ``virtual braids''~\cite{Bardakov:VirtualAndUniversal,
KauffmanLambropoulou:VirtualBraids, BardakovBellingeri:VirtualBraids},
or v-braids, modulo one additional relation; hence, we start with v-braids.

It is simplest to define v-braids in
terms of generators and relations, either algebraically or pictorially.
This can be done in at least two ways -- the easier-at-first but
philosophically less satisfying ``planar'' way, and the harder-to-digest 
but morally more correct ``abstract'' way.\footnote{Compare with a
similar choice that exists in the definition of manifolds, as either
appropriate subsets of some ambient Euclidean spaces (modulo some
equivalences) or as abstract gluings of coordinate patches (modulo some
other equivalences). Here in the ``planar'' approach of
Section~\ref{subsubsec:Planar} we consider v-braids
as ``planar'' objects, and in the ``abstract approach'' of
Section~\ref{subsubsec:Abstract} they are just ``gluings'' of abstract
``crossings'', not drawn anywhere in particular.}

\subsubsection{The ``Planar'' Way} \label{subsubsec:Planar} For a
natural number $n$ set $\glos{\vB_n}$ to be the group generated by
symbols $\glos{\sigma_i}$ ($1\leq i\leq n-1$), called ``crossings''
and graphically represented by an overcrossing $\overcrossing$ ``between
strand $i$ and strand $i+1$'' (with inverse $\undercrossing$)\footnote{We
sometimes refer to $\overcrossing$ as a ``positive crossing'' and to
$\undercrossing$ as a ``negative crossing''.}, and $\glos{s_i}$, called
``virtual crossings'' and graphically represented by a non-crossing,
$\virtualcrossing$, also ``between strand $i$ and strand $i+1$'',
subject to the following relations:

\begin{myitemize}

\item The subgroup of $\vB_n$ generated by the virtual crossings $s_i$
is the symmetric group $\glos{S_n}$, and the $s_i$ correspond to the
transpositions $(i,i+1)$. That is, we have
\begin{equation} \label{eq:sRelations}
  s_i^2=1,
  \qquad s_is_{i+1}s_i = s_{i+1}s_is_{i+1},
  \qquad\text{and if $|i-j|>1$, then}
  \qquad s_is_j=s_js_i.
\end{equation}
In pictures, this is
\begin{equation} \label{eq:sRels}
  \def\i{{$i$}}
  \def\ip{{$i\!+\!1$}}
  \def\ipp{{$i\!+\!2$}}
  \def\j{{$j$}}
  \def\jp{{$j\!+\!1$}}
  \pstex{sRels}
\end{equation}
Note that we read our braids from bottom to top, and that all relations
(and most pictures in this paper) are local: the braids may be bigger
than shown but the parts not shown remain the same throughout a relation.

\item The subgroup of $\vB_n$ generated by the crossings $\sigma_i$ is
the usual braid group $\glos{\uB_n}$, and $\sigma_i$ corresponds to the ``braiding
of strand $i$ over strand $i+1$''. That is, we have
\begin{equation} \label{eq:R3}
  \sigma_i\sigma_{i+1}\sigma_i
    = \sigma_{i+1}\sigma_i\sigma_{i+1},
  \qquad\text{and if $|i-j|>1$ then}
  \qquad \sigma_i\sigma_j=\sigma_j\sigma_i.
\end{equation}
In pictures, dropping the indices, this is
\begin{equation} \label{eq:sigmaRels} \pstex{sigmaRels} \end{equation}
The first of these relations is the ``Reidemeister 3 move''\footnote{The
Reidemeister 2 move is the relation $\sigma_i\sigma_i^{-1}=1$ which is
part of the definition of a group. There is no Reidemeister 1 move in
the theory of braids.} of knot theory. The second is sometimes called
``locality in space''~\cite{Bar-Natan:NAT}.

\item Some ``mixed relations'', that is,
\begin{equation} \label{eq:MixedRelations}
  s_i\sigma^{\pm 1}_{i+1}s_i = s_{i+1}\sigma^{\pm 1}_is_{i+1},
  \qquad\text{and if $|i-j|>1$, then}
  \qquad s_i\sigma_j=\sigma_js_i.
\end{equation}
In pictures, this is
\begin{equation} \label{eq:MixedRels}
  \pstex{MixedRels}
\end{equation}

\end{myitemize}

\begin{remark} \label{rem:Skeleton} The ``skeleton'' of a v-braid $B$
is the set of strands appearing in it, retaining the association
between their beginning and ends but ignoring all the crossing
information. More precisely, it is the permutation induced by
tracing along $B$, and even more precisely it is the image of $B$
via the ``skeleton morphism'' $\glos{\varsigma}\colon\vB_n\to S_n$ defined
by $\varsigma(\sigma_i)=\varsigma(s_i)=s_i$ (or pictorially, by
$\varsigma(\overcrossing)=\varsigma(\virtualcrossing)=\virtualcrossing$).
Thus, the symmetric group $S_n$ is both a subgroup and a quotient group
of $\vB_n$.
\end{remark}

Like there are pure braids to accompany braids, there are pure virtual
braids as well:

\begin{definition} A pure v-braid is a v-braid whose skeleton is the
identity permutation; the group $\glos{\PvB_n}$ of all pure v-braids is
simply the kernel of the skeleton morphism $\varsigma\colon\vB_n\to S_n$.
\end{definition}

We note the short exact sequence of group homomorphisms
\begin{equation} \label{eq:ExcatSeqForPvB}
  1\longrightarrow\PvB_n\xhookrightarrow{\quad}\vB_n
  \overset{\varsigma}{\longrightarrow}S_n
  \longrightarrow 1.
\end{equation}
This short exact sequence splits, with the splitting given by the inclusion
$S_n\hookrightarrow\vB_n$ mentioned above~\eqref{eq:sRelations}.
Therefore, we have that
\begin{equation} \label{eq:vBSemiDirect}
  \vB_n=\PvB_n\rtimes S_n.
\end{equation}

\subsubsection{The ``Abstract'' Way} \label{subsubsec:Abstract}
The relations~\eqref{eq:sRels} and~\eqref{eq:MixedRels} that govern
the behaviour of virtual crossings precisely say that virtual crossings
really are ``virtual'' --- if a piece of strand is routed within a braid
so that there are only virtual crossings around it, it can be rerouted
in any other ``virtual only'' way, provided the ends remain fixed
(this is Kauffman's ``detour move''~\cite{Kauffman:VirtualKnotTheory,
KauffmanLambropoulou:VirtualBraids}). Since a v-braid $B$ is independent
of the routing of virtual pieces of strand, we may as well never supply
this routing information.

\parpic[r]{$\pstex{PvBExample}$}
Thus, for example, a perfectly fair verbal description of the
(pure!) v-braid on the right is ``strand 1 goes over strand 3 by a
positive crossing then likewise positively over strand 2 then negatively
over 3 then 2 goes positively over 1''. We don't need to specify how
strand 1 got to be near strand 3 so it can go over it --- it got there
by means of virtual crossings, and it doesn't matter how. Hence we arrive
at the following ``abstract'' presentation of $\PvB_n$ and $\vB_n$:

\begin{proposition} (E.g.~\cite[Theorems 1 and 2]{Bardakov:VirtualAndUniversal})
\begin{enumerate}
\item The group $\PvB_n$ of pure v-braids
is isomorphic to the group generated by symbols $\glos{\sigma_{ij}}$ for
$1\leq i\neq j\leq n$ (meaning ``strand $i$ crosses over strand $j$
at a positive crossing''\footnote{The inverse, $\sigma_{ij}^{-1}$,
is ``strand $i$ crosses over strand $j$ at a negative crossing''}),
subject to the third Reidemeister move and to locality in space (compare
with~\eqref{eq:R3} and~\eqref{eq:sigmaRels}):

\begin{align*}
  \sigma_{ij}\sigma_{ik}\sigma_{jk} &= \sigma_{jk}\sigma_{ik}\sigma_{ij}
    & \text{whenever}\qquad & |\{i,j,k\}|=3, \\
  \sigma_{ij}\sigma_{kl} &= \sigma_{kl}\sigma_{ij}
     & \text{whenever}\qquad & |\{i,j,k,l\}|=4.
\end{align*}
\item If $\tau\in S_n$, then with the action
$\sigma_{ij}^\tau:=\sigma_{\tau i,\tau j}$ we recover the semi-direct
product decomposition $\vB_n=\PvB_n\rtimes S_n$. \qed
\end{enumerate}
\end{proposition}

\draftcut \subsection{On to w-Braids} \label{subsec:wBraids}
To define w-braids, we break the symmetry between overcrossings and
undercrossings by imposing one of the ``forbidden moves'' in virtual knot
theory, but not the other:
\begin{equation} \label{eq:OvercrossingsCommute}
  \sigma_i\sigma_{i+1}s_i = s_{i+1}\sigma_i\sigma_{i+1},
  \qquad\text{yet}\qquad
  s_i\sigma_{i+1}\sigma_i \neq \sigma_{i+1}\sigma_is_{i+1}.
\end{equation}
Alternatively,
\[ \sigma_{ij}\sigma_{ik} = \sigma_{ik}\sigma_{ij},
  \qquad\text{yet}\qquad
  \sigma_{ik}\sigma_{jk} \neq \sigma_{jk}\sigma_{ik}.
\]
In pictures, this is
\begin{equation} \label{eq:OC} \pstex{OCUC} \end{equation}

The relation we have just imposed may be called the ``unforbidden
relation'', or, perhaps more appropriately, the ``overcrossings commute''
relation, abbreviated \glost{OC}. Ignoring the non-crossings\footnote{Why this is
appropriate was explained in the previous section.} $\virtualcrossing$,
the OC relation says that it is the same if strand $i$ first crosses
over strand $j$ and then over strand $k$, or if it first crosses over
strand $k$ and then over strand $j$. The ``undercrossings commute''
relation \glost{UC}, the one we do not impose
in~\eqref{eq:OvercrossingsCommute}, would say the same except with
``under'' replacing ``over''.

\begin{definition} The group of w-braids is $\glos{\wB_n}:=\vB_n/OC$. Note
that $\varsigma$ descends to $\wB_n$, and hence we can define the group $\glos{\PwB_n}$ of
pure w-braids to be the kernel of the map $\varsigma\colon\wB_n\to S_n$. We
still have a split exact sequence as in~\eqref{eq:ExcatSeqForPvB} and
a thus, a semi-direct product decomposition $\wB_n=\PwB_n\rtimes S_n$.
\end{definition}

\begin{exercise} Show that the OC relation is equivalent to the relation
\[
  \sigma_i^{-1}s_{i+1}\sigma_i = \sigma_{i+1}s_i\sigma_{i+1}^{-1}
  \qquad\text{or}\qquad
  \parbox[m]{1.5in}{\input figs/AltOC.pstex_t }
\]
\end{exercise}

While for most of this paper the pictorial / algebraic definition of w-braids
(and other w-knotted objects) will suffice, we ought to describe at least
briefly a few further interpretations of $\wB_n$:

\subsubsection{The Group of Flying Rings} \label{subsubsec:FlyingRings}
Let \glos{X_n} be the space of all placements of $n$ numbered disjoint
geometric circles in $\bbR^3$, such that all circles are parallel to
the $xy$ plane. Such placements will be called horizontal\footnote{
For the group of non-horizontal flying rings see
Section \ref{subsubsec:NonHorRings}.}. A horizontal
placement is determined by the centres in $\bbR^3$ of the $n$ circles
and by $n$ radii, so $\dim X_n=3n+n=4n$. The permutation group $S_n$
acts on $X_n$ by permuting the circles, and one may think of the quotient
$\glos{\tilde{X}_n}:=X_n/S_n$ as the space of all horizontal
placements of $n$ unmarked circles in $\bbR^3$. The fundamental group
$\pi_1(\tilde{X}_n)$ is a group of paths traced by $n$ disjoint horizontal
circles (modulo homotopy), so it is fair to think of it as ``the group
of flying rings''.

\begin{theorem} The group of pure w-braids $\PwB_n$ is isomorphic to the group
of flying rings $\pi_1(X_n)$. The group $\wB_n$ is isomorphic to the group
of unmarked flying rings $\pi_1(\tilde{X}_n)$.
\end{theorem}

For the proof of this theorem,
see~\cite{Goldsmith:MotionGroups, Satoh:RibbonTorusKnots} and
especially~\cite[Proposition~3.3]{BrendleHatcher:RingsAndWickets}. Here
we will content ourselves with pictures describing the images of the
generators of $\wB_n$ in $\pi_1(\tilde{X}_n)$ and a few comments:

\[ \input figs/FlyingRings.pstex_t \]

Thus, we map the permutation $s_i$ to the movie clip in which ring
number $i$ trades places with ring number $i+1$ by having the two
fly around each other. This acrobatic feat is performed in $\bbR^3$
and it does not matter if ring number $i$ goes ``above'' or ``below''
or ``left'' or ``right'' of ring number $i+1$ when they trade places,
as all of these possibilities are homotopic. More interestingly, we
map the braiding $\sigma_i$ to the movie clip in which ring $i+1$
shrinks a bit and flies through ring $i$. It is a worthwhile
exercise for the reader to verify that the relations in the definition
of $\wB_n$ become homotopies of movie clips. Of these relations it
is most interesting to see why the ``overcrossings commute'' relation
$\sigma_i\sigma_{i+1}s_i = s_{i+1}\sigma_i\sigma_{i+1}$ holds, yet the
``undercrossings commute'' relation $\sigma^{-1}_i\sigma^{-1}_{i+1}s_i =
s_{i+1}\sigma^{-1}_i\sigma^{-1}_{i+1}$ doesn't.

\begin{exercise}\label{ex:swBn}
To be perfectly precise, we have to
specify the fly-through direction. In our notation, $\sigma_i$ means
that the ring corresponding to the strand going under (in the local picture 
for $\sigma_i$) approaches from below the bigger
ring representing the strand going over, then flies through it and exists
above.  For $\sigma_i^{-1}$ we are ``playing the movie backwards'', i.e.,
the ring of the strand going under comes from above and exits below the ring
of the ``over'' strand.

Let ``the signed $w$ braid group'', $\swB_n$, be the group
of horizontal flying rings where both fly-through 
directions are allowed. This introduces a ``sign'' for
each crossing $\sigma_i$:
\begin{center}
 \input figs/FlyingRings2.pstex_t
\end{center}
In other words, $\swB_n$ is generated by $s_i$, 
$\sigma_{i+}$ and $\sigma_{i-}$, for $i=1,...,n-1$. Check that
$\sigma_{i-}=s_i\sigma_{i+}^{-1}s_i$ in $\swB_n$, and this, along with
the other obvious relations implies $\swB_n \cong \wB_n$.
\end{exercise}

\subsubsection{Certain Ribbon Tubes in $\bbR^4$} \label{subsubsec:ribbon}
With
time as the added dimension, a flying ring in $\bbR^3$ traces a tube
(an annulus) in $\bbR^4$, as shown in the picture below:
\[ \input figs/RibbonTubes.pstex_t \]
Note that we adopt here the drawing conventions of Carter and
Saito~\cite{CarterSaito:KnottedSurfaces} --- we draw surfaces as if they
were projected from $\bbR^4$ to $\bbR^3$, and we cut them open whenever
they are ``hidden'' by something with a higher fourth coordinate.

Note
also that the tubes we get in $\bbR^4$ always bound natural 3D
``solids'' --- their ``insides'', in the pictures above. These solids
are disjoint in the case of $s_i$ and have a very specific kind
of intersection in the case of $\sigma_i$ --- these are transverse
intersections with no triple points, and their inverse images are a
meridional disk on the ``thin'' solid tube and an interior disk on the
``thick'' one. By analogy with the case of ribbon knots and ribbon
singularities in $\bbR^3$ (e.g.~\cite[Chapter V]{Kauffman:OnKnots}) and
following Satoh~\cite{Satoh:RibbonTorusKnots}, we call these kinds of
intersections of solids in $\bbR^4$ ``ribbon singularities'' and thus, our
tubes in $\bbR^4$ are always ``ribbon tubes''.

\subsubsection{Basis Conjugating Automorphisms of $F_n$}
\label{subsubsec:McCool}
Let
$\glos{F_n}$ be the free (non-abelian) group with generators
$\glos{\xi_1,\ldots,\xi_n}$. Artin's theorem (Theorems 15 and 16
of~\cite{Artin:TheoryOfBraids}) says that the (usual) braid
group $\uB_n$ (equivalently, the subgroup of $\wB_n$ generated by
the $\sigma_i$) has a faithful right action on $F_n$. In other
words, $\uB_n$ is isomorphic to a subgroup $H$ of $\Autop(F_n)$
(the group of automorphisms of $F_n$ with opposite multiplication, i.e.,
$\psi_1\psi_2:=\psi_2\circ\psi_1$). Precisely, using $(\xi,
B)\mapsto\xi\glos{\sslash}B$ to denote the right action of $\Autop(F_n)$ on
$F_n$, the subgroup $H$ consists of those automorphisms $B\colon F_n\to F_n$
of $F_n$ that satisfy the following two conditions:
\begin{enumerate}
\item $B$ maps any generator $\xi_i$ to a
  conjugate of a generator (possibly different). That is, there is a
  permutation $\beta\in S_n$ and elements $a_i\in F_n$ such that, for every $i$,
  \begin{equation} \label{eq:BasisConjugating}
    \xi_i \sslash B = a_i^{-1}\xi_{\beta (i)}a_i.
  \end{equation}
\item $B$ fixes the ordered product of the generators of $F_n$,
  \[ \xi_1\xi_2\cdots \xi_n \sslash B = \xi_1\xi_2\cdots \xi_n. \]
\end{enumerate}

McCool's theorem\footnote{Strictly speaking, the main theorem
of~\cite{McCool:BasisConjugating} is about $\PwB_n$, yet it can easily be
restated for $\wB_n$.}~\cite{McCool:BasisConjugating} says that almost the same statement 
holds true\footnote{Though see Warning~\ref{warn:NoArtin}.} for the bigger group $\wB_n$:
namely, $\wB_n$ is isomorphic to the subgroup of $\Autop(F_n)$ consisting of
automorphisms satisfying only the first condition above.
So $\wB_n$ is precisely the group
of ``basis-conjugating'' automorphisms of the free group $F_n$,
the group of those automorphisms which map any ``basis element''
in $\{\xi_1,\ldots,\xi_n\}$ to a conjugate of a (possibly different)
basis element.

The relevant action is explicitly defined on the generators of $\wB_n$
and $F_n$ as follows (we state how each generator of $\wB_n$ acts on 
each generator of $F_n$, in each case omitting the generators of $F_n$ 
which are fixed under the action):
\begin{equation} \label{eq:ExplicitPsi}
  (\xi_i, \xi_{i+1})\sslash s_i = (\xi_{i+1}, \xi_i),
  \qquad
  (\xi_i, \xi_{i+1})\sslash \sigma_i = (\xi_{i+1},
    \xi_{i+1}\xi_i\xi_{i+1}^{-1}),
  \qquad
  \xi_j\sslash \sigma_{ij} = \xi_i\xi_j\xi_i^{-1}.
\end{equation} 
It is a worthwhile exercise to verify that $\sslash$ respects the
relations in the definition of $\wB_n$ and that the permutation $\beta$
in~\eqref{eq:BasisConjugating} is the skeleton $\varsigma(B)$.

There is a more conceptual description of $\sslash$, in terms of the
structure of $\wB_{n+1}$. Consider the inclusions
\begin{equation} \label{eq:inclusions}
  \wB_n \xhookrightarrow{\iota} \wB_{n+1} \xhookleftarrow{i_u} F_n.
\end{equation}


Here $\glos{\iota}$ is the inclusion of $\wB_n$ into $\wB_{n+1}$ by adding an
inert $(n+1)$st strand (it is injective as it has a well-defined
one sided inverse -- the deletion of the $(n+1)$st strand). 

\parpic[r]{$\pstex{xi}$}
The inclusion $\glos{i_u}$ of the free group $F_n$ into $\wB_{n+1}$ is defined by
$i_u(\xi_i):=\sigma_{i,n+1}$.  The image $i_u(F_n)\subset\wB_{n+1}$ is the
set of all w-braids whose first $n$ strands are straight and vertical,
and whose $(n+1)$-st strand wanders among the first $n$ strands mostly
virtually (i.e., mostly using virtual crossings), occasionally slipping
under one of those $n$ strands, but never going over anything. 
It is easier to see that
this is indeed injective using the
``flying rings'' picture of Section~\ref{subsubsec:FlyingRings}. The image
$i_u(F_n)\subset\wB_{n+1}$ can be interpreted as the fundamental group
of the complement in $\bbR^3$ of $n$ stationary rings (which is indeed
$F_n$) --- in $i_u(F_n)$ the only ring in motion is the last, and it only
goes under, or ``through'', other rings, so it can be replaced by a point
object whose path is an element of the fundamental group. The injectivity
of $i_u$ follows from this geometric picture.

\parpic[r]{$\pstex{Bgamma}$} \picskip{4}
One may explicitly verify that $i_u(F_n)$ is normalized by $\iota(\wB_n)$
in $\wB_{n+1}$ (that is, the set $i_u(F_n)$ is preserved by conjugation
by elements of $\iota(\wB_n)$). Thus, the following definition (also shown
as a picture on the right) makes sense, for \linebreak $B\in\wB_n\subset\wB_{n+1}$
and for $\gamma\in F_n\subset\wB_{n+1}$:

\begin{equation} \label{eq:ConceptualPsi}
  \gamma\sslash B := i_u^{-1}(B^{-1}\gamma B)
\end{equation}

It is a worthwhile exercise to recover the explicit formulae
in~\eqref{eq:ExplicitPsi} from the above definition.

\begin{warning} \label{warn:NoArtin} People familiar with the Artin story for
ordinary braids should be warned that even though $\wB_n$ acts on $F_n$ and
the action is induced from the inclusions in~\eqref{eq:inclusions} in much
the same way as the Artin action is induced by inclusions $\uB_n
\xhookrightarrow{\iota} \uB_{n+1} \xhookleftarrow{i} F_n$, there are also
some differences, and some further warnings apply:
\begin{myitemize}
\item In the ordinary Artin story, $i(F_n)$ is the set of braids in
$\uB_{n+1}$ whose first $n$ strands are unbraided (that is, whose image in
$\uB_n$ via ``dropping the last strand'' is the identity). This is not true
for w-braids. For w-braids, in $i_u(F_n)$ the last strand always goes
``under'' all other strands (or just virtually crosses them), but never
``over''.
\item Thus, unlike the isomorphism $\PuB_{n+1}\cong \PuB_n\ltimes F_n$,
it is not true that $\PwB_{n+1}$ is isomorphic to $\PwB_n\ltimes F_n$.
\item The OC relation imposed in $\wB$ breaks
the symmetry between overcrossings and undercrossings. Thus, let
$i_o\colon F_n\to\wB_n$ be the ``opposite'' of $i_u$, mapping into braids in
which the last strand is always ``over'' or virtual. Then $i_o$ is not
injective (its image is in fact abelian) and its image is not normalized
by $\iota(\wB_n)$. So there is no ``second'' action of $\wB_n$ on $F_n$
defined using $i_o$.
\item For v-braids, both $i_u$ and $i_o$ are injective and there are two
actions of $\vB_n$ on $F_n$ --- one defined by first projecting into
w-braids, and the other defined by first projecting into v-braids modulo
``undercrossings commute''. Yet v-braids contain more information than
these two actions can see. The ``Kishino'' v-braid below, for example,
is visibly trivial if either overcrossings or undercrossings are made to
commute, yet by computing its Kauffman bracket we know it is non-trivial
as a v-braid~\cite[``The Kishino Braid'']{WKO}:
\[
  \pstex{KishinoBraid} \quad \left(\parbox{1.6in}{\footnotesize
    The commutator $ab^{-1}a^{-1}b$ of v-braids $a,b$ annihilated by OC/UC,
    respectively, with a minor cancellation.
  }\right)
\]
\end{myitemize}
\end{warning}

\begin{problem} \label{prob:wCombing}
Are $\PvB_n$ and $\PwB_n$ semi-direct products of free groups?
For $\PuB_n$, this
is the well-known ``combing of braids'' and it follows from $\PuB_n\cong
\PuB_{n-1}\ltimes F_{n-1}$ and induction.
\end{problem}

\begin{remark} \label{rem:GutierrezKrstic}
Note that Guti\'errez and Krsti\'c~\cite{GutierrezKrstic:NormalForms}
have found ``normal forms'' for the elements of $\PwB_n$, yet they
do not decide whether $\PwB_n$ is ``automatic'' in the sense
of~\cite{Epstein:WordProcessing}.
\end{remark}

\draftcut \subsection{Finite Type Invariants of v-Braids and w-Braids}
\label{subsec:FT4Braids}

Just as we had two definitions for v-braids (and thus, for w-braids)
in Section~\ref{subsec:VirtualBraids}, we will give two 
equivalent developments of the theory of finite type invariants of
v-braids and w-braids --- a pictorial/topological version in
Section~\ref{subsubsec:FTPictorial}, and a more abstract algebraic version
in Section~\ref{subsubsec:FTAlgebraic}.

\subsubsection{Finite Type Invariants, the Pictorial Approach}
\label{subsubsec:FTPictorial}

In
the standard theory of finite type invariants of knots (also known as
Vassiliev or Goussarov-Vassiliev invariants)~\cite{Vassiliev:CohKnot, Goussarov:nEquivalence,
Bar-Natan:OnVassiliev, Bar-Natan:EMP}
one progresses from the definition of finite type via iterated
differences to chord diagrams and weight systems, to $4T$ (and other)
relations, to the definition of universal finite type invariants,
and beyond. The exact same progression (with different objects
playing similar roles, and sometimes, when yet insufficiently
studied, with the last step or two missing) is also seen in the theories
of finite type invariants of braids~\cite{Bar-Natan:Braids},
3-manifolds~\cite{Ohtsuki:IntegralHomology, LeMurakamiOhtsuki:Universal,
Le:UniversalIHS}, virtual knots~\cite{GoussarovPolyakViro:VirtualKnots,
Polyak:ArrowDiagrams} and of several other classes of objects. We thus
assume that the reader has familiarity with these basic ideas, and we
only indicate briefly how they are implemented in the case of v-braids
and w-braids.

\begin{figure}
\[ \input figs/Dvh1.pstex_t \]
  \caption{
    On the left, a 3-singular v-braid and its corresponding 3-arrow
    diagram. A self-explanatory algebraic notation for this arrow
    diagram is $(\glos{a_{12}a_{14}a_{43}},\,3412)$. 
    Note that we regard arrow diagrams as graph-theoretic
    objects, and hence, the two arrow diagrams on the right, whose
    underlying graphs are the same, are regarded as equal. In
    algebraic notation this means that we always impose the relation
    $a_{ij}a_{kl}=a_{kl}a_{ij}$ when the indices $i$, $j$, $k$, and $l$
    are all distinct.
  } \label{fig:Dvh1}
\end{figure}

Much
like the formula $\doublepoint\to\overcrossing-\undercrossing$ of
Vassiliev-Goussarov fame, given a v-braid invariant $\glos{V}\colon
\vB_n\to A$ valued in some abelian group $A$, we extend it to
``singular'' v-braids, i.e., braids that contain ``semi-virtual crossings''
like $\glos{\semivirtualover}$ and $\glos{\semivirtualunder}$ using the
formulae $V(\semivirtualover):=V(\overcrossing)-V(\virtualcrossing)$
and $V(\semivirtualunder):=V(\undercrossing)-V(\virtualcrossing)$
(see~\cite{GoussarovPolyakViro:VirtualKnots, Polyak:ArrowDiagrams,
Bar-NatanHalachevaLeungRoukema:v-Dims}). We
say that ``$V$ is of type $m$'' if its extension vanishes on singular
v-braids having more than $m$ semi-virtual crossings. Up to invariants
of lower type, an invariant of type $m$ is determined by its ``weight
system'', which is a functional $W=\glos{W_m}(V)$ defined on ``$m$-singular
v-braids modulo $\overcrossing=\virtualcrossing=\undercrossing$''. Let
us denote the vector space of all formal linear combinations of such
equivalence classes by $\glos{\calG_m}\calD^v_n$. Much as $m$-singular knots
modulo $\overcrossing=\undercrossing$ can be identified with chord
diagrams, the basis elements of $\calG_m\calD^v_n$ can be identified with
pairs $(D,\beta)$, where $D$ is a horizontal arrow diagram and $\beta$
is a ``skeleton permutation'', see Figure~\ref{fig:Dvh1}.

We assemble the spaces $\calG_m\calD^v_n$ together to form a single
graded space, $\glos{\calD^v_n}:=\oplus_{m=0}^\infty\calG_m\calD^v_n$. Note that
throughout this paper, whenever we write an infinite direct sum,
we automatically complete it. Thus, in $\calD^v_n$ we allow infinite sums
with one term in each homogeneous piece $\calG_m\calD^v_n$, in particular,
exponential-like sums will be heavily used.

\begin{figure}
\[ \input figs/6T.pstex_t \]
\[
  a_{ij}a_{ik}+a_{ij}a_{jk}+a_{ik}a_{jk}
  = a_{ik}a_{ij}+a_{jk}a_{ij}+a_{jk}a_{ik}
\]
\[
  \text{or}\qquad
  [a_{ij}, a_{ik}] + [a_{ij}, a_{jk}] + [a_{ik}, a_{jk}] = 0
\]
\caption{The $6T$ relation. Standard knot theoretic conventions apply ---
only the relevant parts of each diagram are shown; in reality each diagram
may have further vertical strands and horizontal arrows, provided the
extras are the same in all 6 diagrams. Also, the vertical strands are in no
particular order --- other valid $6T$ relations are obtained when those
strands are permuted in other ways.} \label{fig:6T}
\end{figure}

\begin{figure}
\[ \begin{array}{ccc}
  \input figs/TC.pstex_t & \qquad & \input figs/4TArrow.pstex_t \\
  a_{ij}a_{ik} = a_{ik}a_{ij} &&
  a_{ij}a_{jk} + a_{ik}a_{jk} = a_{jk}a_{ij} + a_{jk}a_{ik} \\
  \text{or} \quad [a_{ij}, a_{ik}] = 0 &&
  \text{or} \quad [a_{ij} + a_{ik}, a_{jk}] = 0
\end{array} \]
\caption{The TC and the $\protect\aft$ relations.} \label{fig:TCand4T}
\end{figure}

In the standard finite-type theory for knots, weight
systems always satisfy the $4T$ relation, and are therefore
functionals on $\calA:=\calD/4T$. Likewise, in the case of
v-braids, weight systems satisfy the ``$\glos{6T}$ relation''
of~\cite{GoussarovPolyakViro:VirtualKnots, Polyak:ArrowDiagrams,
Bar-NatanHalachevaLeungRoukema:v-Dims},
shown in Figure~\ref{fig:6T}, and are therefore functionals on
$\glos{\calA^v_n}:=\calD^v_n/6T$. In the case of w-braids, the OC 
relation~\eqref{eq:OvercrossingsCommute} implies the
``tails commute'' (\glost{TC}) relation on the level of arrow diagrams,
and in the presence of the TC relation, two of the terms in the $6T$
relation drop out, and what remains is the ``$\glos{\aft}$'' relation. These
relations are shown in Figure~\ref{fig:TCand4T}. Thus, weight systems
of finite type invariants of w-braids are linear functionals on
$\glos{\calA^w_n}:=\calD^v_n/TC,\aft$.

The next question that arises is whether we have already found {\em
all} the relations that weight systems always satisfy. More precisely,
given a degree $m$ linear functional on $\calA^v_n=\calD^v_n/6T$ (or
on $\calA^w_n=\calD^v_n/TC,\aft$), is it always the weight system
of some type $m$ invariant $V$ of v-braids (or w-braids)? As in every
other theory of finite type invariants, the answer to this question
is affirmative if and only if there exists a ``universal finite type
invariant'' (or simply, an ``expansion'') of v-braids (or w-braids),
defined as follows:

\begin{definition} \label{def:vwbraidexpansion} An expansion
for v-braids (or w-braids) is an invariant $Z\colon \vB_n\to\calA^v_n$
(or $Z\colon \wB_n\to\calA^w_n$) satisfying the following ``universality
condition'':
\begin{itemize}
\item If $B$ is an $m$-singular v-braid (or w-braid) and
$D\in\calG_m\calD^v_n$ is its underlying arrow diagram as in
Figure~\ref{fig:Dvh1}, then
\[ Z(B)=D+(\text{terms of degree\,}>m). \]
\end{itemize}
\end{definition}

Indeed if $Z$ is an expansion and $W\in\calG_m\calA^\star$,\footnote{$\calA^\star$
here denotes either $\calA^v_n$ or $\calA^w_n$, or in fact, any of many
similar spaces that we will discuss later on.} the universality condition
implies that $W\circ Z$ is a finite type invariant whose weight system
is $W$. To go the other way, if $(D_i)$ is a basis of $\calA$ consisting
of homogeneous elements, if $(W_i)$ is the dual basis of $\calA^\star$ and
$(V_i)$ are finite type invariants whose weight systems are the $W_i$,
then $Z(B):=\sum_iD_iV_i(B)$ defines an expansion.

In general, constructing a universal finite type invariant is a
hard task. For knots, one uses either the Kontsevich integral or
perturbative Chern-Simons theory (also known as ``configuration
space integrals''~\cite{BottTaubes:SelfLinking} or ``tinker-toy
towers''~\cite{Thurston:IntegralExpressions}) or the rather fancy
algebraic theory of ``Drinfel'd associators'' (a summary of all those
approaches is in~\cite{Bar-NatanStoimenow:Fundamental}). For homology
spheres, this is the ``LMO invariant''~\cite{LeMurakamiOhtsuki:Universal,
Le:UniversalIHS} (also the ``\AA{}rhus
integral''~\cite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}). For
v-braids, an expansion exists by a difficult result of
P.~Lee,~\cite{Lee:VirtualIsQuadratic}. In contrast, as we shall see
below, the construction of an expansion for w-braids is quite easy.

\subsubsection{Finite Type Invariants, the Algebraic Approach}
\label{subsubsec:FTAlgebraic}
For any group $G$, one can form the group algebra ${\mathbb Q}G$ by
allowing formal linear combinations of group elements and extending
multiplication linearly, where $\mathbb Q$ is the field of rational
numbers\footnote{The definitions in this subsection make sense over $\bbZ$
as well, but the main result of the next subsection requires a field of
characteristic $0$. For simplicity of notation we stick with $\bbQ$.}.
The {\it augmentation ideal} is the ideal generated by
differences, or equivalently, the set of linear combinations of group
elements whose coefficients sum to zero:
\[ \glos{\calI} := \left\{\sum_{i=1}^k a_ig_i\colon
  a_i \in {\mathbb Q}, g_i \in G, \sum_{i=1}^k a_i=0\right\}.
\]
Powers of the augmentation ideal provide a filtration of the group
algebra. We denote by $\glos{\calA(G)}:= \bigoplus_{m\geq 0} \calI^m/\calI^{m+1}$
the associated graded space corresponding to this filtration.

\begin{definition}\label{def:grpexpansion} An
expansion for the group
$G$ is a map $Z\colon G \to \calA(G)$, such that the linear extension
$Z\colon  {\mathbb Q}G \to \calA(G)$ is filtration preserving and
the induced map $$\gr Z\colon  (\gr {\mathbb Q}G=\calA(G)) \to (\gr
\calA(G)=\calA(G))$$ is the identity. An equivalent way to phrase this
is that the degree $m$ piece of $Z$ restricted to $\calI^m$ is the
projection onto $\calI^m/\calI^{m+1}$.

\begin{exercise}\label{ex:BraidsAlgApproach}
Verify that for the groups $\PvB_n$ and $\PwB_n$ the m-th power of the
augmentation ideal coincides with the span of all resolutions of
$m$-singular $v$- or $w$-braids (by a resolution we mean the formal
linear combination where each semivirtual crossing is replaced by
the appropriate difference of a virtual and a regular crossing, as in Figure \ref{fig:Dvh1}). Then
check that the notion of expansion defined above is the same as that of
Definition \ref{def:vwbraidexpansion}, restricted to pure braids.
\end{exercise}

Finally, note the functorial nature of the construction above. What we
have described is a functor $\grs$ from the category of groups to the
category of graded algebras which assigns to each group $G$ the graded
algebra $\calA(G)$. To each homomorphism $\phi\colon  G \to H$, $\grs$
assigns the induced map
\[ \gr \phi\colon  (\calA(G)=\gr {\mathbb Q}G)
  \to (\calA(H)= \gr {\mathbb Q}H).
\]

\end{definition}


\draftcut \subsection{Expansions for w-Braids}\label{subsec:wBraidExpansion}

The space $\calA^w_n$ of arrow diagrams on $n$ strands is an associative
algebra in an obvious manner: if the permutations underlying two arrow
diagrams are the identity permutations, then we simply juxtapose the
diagrams.  Otherwise we ``slide'' arrows through permutations in the
obvious manner --- if $\tau$ is a permutation, we declare that $\tau
a_{(\tau i)(\tau j)}=a_{ij}\tau$. Instead of seeking an expansion
$\wB_n\to\calA^w_n$, we set the bar a little higher and seek a
``homomorphic expansion'', defined as follows:

\begin{definition} \label{def:Universallity} A homomorphic expansion
$Z\colon \wB_n\to\calA^w_n$ is an expansion that carries products in $\wB_n$
to products in $\calA^w_n$.
\end{definition}

To find a homomorphic expansion, we just need to define it
on the generators of $\wB_n$ and verify that it satisfies the
relations defining $\wB_n$ and the universality condition.
Following~\cite[Section~5.3]{BerceanuPapadima:BraidPermutation}
and~\cite[Section~8.1]{AlekseevTorossian:KashiwaraVergne} we set
$Z(\virtualcrossing)=\virtualcrossing$ (that is, a transposition in
$\wB_n$ gets mapped to the same transposition in $\calA^w_n$, adding no
arrows) and $Z(\overcrossing)=\exp(\rightarrowdiagram)\virtualcrossing$.
(Recall that we work in the degree completion.)
This last formula is important so deserves to be magnified, explained
and replaced by some new notation:

\begin{equation} \label{eq:reservoir}
  Z\left(\!\mathsize{\Huge}{\overcrossing}\!\right)\! =
  \exp\left(\!\mathsize{\Huge}{\rightarrowdiagram}\!\right)
    \cdot\mathsize{\Huge}{\virtualcrossing}
  = \pstex{ZIsExp}+\ldots =: \pstex{ArrowReservoir}.
\end{equation}

Thus the new notation $\overset{e^a}{\longrightarrow}$ stands
for an ``exponential reservoir'' of parallel arrows, much like
$e^a=1+a+aa/2+aaa/3!+\ldots$ is a ``reservoir'' of $a$'s. With
the obvious interpretation for $\overset{e^{-a}}{\longrightarrow}$
(that is, the $-$ sign indicates that the terms should have alternating signs,
as in $e^{-a}=1-a+a^2/2-a^3/3!+\ldots$), the second Reidemeister move
$\overcrossing\undercrossing=1$ forces that we set
\[ Z\left(\mathsize{\Huge}{\undercrossing}\right) =
  \mathsize{\Huge}{\virtualcrossing}
  \cdot\exp\left(-\mathsize{\Huge}{\rightarrowdiagram}\right)
  = \pstex{NegReservoir1} = \pstex{NegReservoir2}.
\]

\begin{theorem} \label{thm:RInvariance} The above formulae define
an invariant $Z\colon \wB_n\to\calA^w_n$ (that is, $Z$ satisfies all the
defining relations of $\wB_n$). The resulting $Z$ is a homomorphic
expansion (that is, it satisfies the universality property of
Definition~\ref{def:Universallity}).
\end{theorem}

\begin{proof} Following~\cite{BerceanuPapadima:BraidPermutation,
AlekseevTorossian:KashiwaraVergne}: for the invariance of $Z$, the
only interesting relations to check are the Reidemeister 3 relation
of~\eqref{eq:sigmaRels} and the OC relation
of~\eqref{eq:OC}. For Reidemeister 3, we have
\[ \pstex{R3Left}
  = e^{a_{12}}e^{a_{13}}e^{a_{23}}\tau
  \overset{1}{=} e^{a_{12}+a_{13}}e^{a_{23}}\tau
  \overset{2}{=} e^{a_{12}+a_{13}+a_{23}}\tau,
\]
where $\tau$ is the permutation $321$ and equality 1 holds because
$[a_{12},a_{13}]=0$ by a TC relation and equality 2 holds
because $[a_{12}+a_{13}, a_{23}]=0$ by a $\aft$ relation.
Likewise, again using TC and $\aft$,
\[ \pstex{R3Right}
  = e^{a_{23}}e^{a_{13}}e^{a_{12}}\tau
  = e^{a_{23}}e^{a_{13}+a_{12}}\tau
  = e^{a_{23}+a_{13}+a_{12}}\tau,
\]
and so Reidemeister 3 holds. An even simpler proof using just the TC 
relation shows that the OC relation also holds.
Finally, since $Z$ is homomorphic, it is enough to check the universality
property at degree $1$, where it is very easy:
\[ Z\left(\mathsize{\Huge}{\semivirtualover}\right) =
  \exp\left(\mathsize{\Huge}{\rightarrowdiagram}\right)
    \cdot\mathsize{\Huge}{\virtualcrossing}
    - \mathsize{\Huge}{\virtualcrossing}
  = \mathsize{\Huge}{\rightarrowdiagram}\cdot\mathsize{\Huge}{\virtualcrossing}
    + (\text{terms of degree\,}>1).
\]
A similar computation manages the $\semivirtualunder$ case. \qed
\end{proof}

\begin{remark} \label{rem:YangBaxter} Note that the main ingredient
of the above proof was to show that \linebreak $\glos{R}:=Z(\sigma_{12})=e^{a_{12}}$
satisfies the famed Yang-Baxter equation,
\[ R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12}, \]
where $R^{ij}$ means ``place $R$ on strands $i$ and $j$''.
\end{remark}

\draftcut
\subsection{Some Further Comments} \label{subsec:bcomments}
\subsubsection{Compatibility with Braid Operations}
  \label{subsubsec:BraidCompatibility}
As with
any new gadget, we would like to know how compatible the expansion
$Z$ of the previous section is with the gadgets we already have; namely,
with various operations that are available on w-braids and with the action
of w-braids on the free group $F_n$, see Section~\ref{subsubsec:McCool}.

\parpic[r]{$\xymatrix{
  \wB_n \ar[r]^\theta \ar[d]_Z   & \wB_n \ar[d]^Z    \\
  \calA^w_n \ar[r]_\theta        & \calA^w_n
    \ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{$Z$ is Compatible with Braid Inversion} \label{par:theta}
Let $\theta$ denote both the ``braid inversion'' operation
$\glos{\theta}\colon \wB_n\to\wB_n$ defined by $B\mapsto B^{-1}$ and
the ``antipode'' anti-automorphism $\theta\colon \calA^w_n\to\calA^w_n$
defined by mapping permutations to their inverses and arrows to their
negatives (that is, $a_{ij}\mapsto-a_{ij}$). Then the diagram on the
right commutes.

\pagebreak[2]

\parpic[r]{$\xymatrix{
  \wB_n \ar[r]^<>(0.5)\Delta \ar[d]_Z   & \wB_n\times\wB_n \ar[d]^{Z\times Z} \\
  \calA^w_n \ar[r]_<>(0.5)\Delta        & \calA^w_n\otimes\calA^w_n
    \ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Braid Cloning and the Group-Like Property} \label{par:Delta}
Let $\glos{\Delta}$ denote both the ``braid cloning''
operation $\Delta\colon \wB_n\to\wB_n\times\wB_n$ defined by
$B\mapsto (B,B)$ and the ``co-product'' algebra morphism \linebreak
$\Delta\colon \calA^w_n\to\calA^w_n\otimes\calA^w_n$ defined by cloning
permutations (that is, $\tau\mapsto\tau\otimes\tau$) and by treating
arrows as primitives (that is, \linebreak $a_{ij}\mapsto a_{ij}\otimes 1+1\otimes
a_{ij}$). Then the diagram on the right commutes. In formulae, this is
$\Delta(Z(B))=Z(B)\otimes Z(B)$, which is the statement ``$Z(B)$ is
group-like''.

\parpic[r]{$\xymatrix{
  \wB_n \ar[r]^<>(0.5)\iota \ar[d]_Z	& \wB_{n+1} \ar[d]^Z	\\
  \calA^w_n \ar[r]_<>(0.5)\iota	& \calA^w_{n+1}
    \ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Strand Insertions} \label{par:iota}
Let $\iota\colon \wB_n\to\wB_{n+1}$ be an operation of ``inert strand
insertion''. Given $B\in\wB_n$, the resulting $\iota B\in\wB_{n+1}$
will be $B$ with one strand $S$ added at some location chosen in
advance --- to the left of all existing strands, or to the right, or
starting from between the 3rd and the 4th strand of $B$ and ending
between the 6th and the 7th strand of $B$; when adding $S$, add it
``inert'', so that all crossings on it are virtual (this is well
defined). There is a corresponding inert strand addition operation
$\iota\colon \calA^w_n\to\calA^w_{n+1}$, obtained by adding a strand at the
same location as for the original $\iota$ and adding no arrows. It is
easy to check that $Z$ is compatible with $\iota$; namely, that the
diagram on the right is commutative.

\parpic[r]{$\xymatrix{
  \wB_n \ar[r]^<>(0.5){d_k} \ar[d]_Z   & \wB_{n-1} \ar[d]^Z    \\
  \calA^w_n \ar[r]_<>(0.5){d_k}        & \calA^w_{n-1}
    \ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Strand Deletions} \label{par:deletions} Given $1 \leq k \leq n$, 
let $\glos{d_k}\colon \wB_n\to\wB_{n-1}$ be the operation of
``removing the $k$th strand''.  This operation induces a homonymous
operation $d_k\colon \calA^w_n\to\calA^w_{n-1}$: if $D\in\calA^w_n$ is an
arrow diagram, then $d_kD$ is $D$ with its $k$th strand removed if no arrows
in $D$ start or end on the $k$th strand, and it is $0$ otherwise. It
is easy to check that $Z$ is compatible with $d_k$; namely, that the
diagram on the right is commutative.\footnote{In \cite{Bar-NatanDancso:WKO2} we'll 
say that ``$d_k\colon \wB_n\to\wB_{n-1}$''
is an algebraic structure made of two spaces ($\wB_n$ and $\wB_{n-1}$),
two binary operations (braid composition in $\wB_n$ and in $\wB_{n-1}$),
and one unary operation, $d_k$. After applying $\grs$ we get the
algebraic structure $d_k\colon \calA^w_n\to\calA^w_{n-1}$ with $d_k$
as described above, and an alternative way of stating our assertion is
to say that $Z$ is a morphism of algebraic structures. A similar remark
applies (sometimes in the negative form) to the other operations discussed
in this section.}

\parpic[r]{$\xymatrix{
  F_n \ar@{}[r]|{\mathsize{\Huge}{\actsonright}} \ar[d]_Z & \wB_n \ar[d]^Z \\
  \FA_n \ar@{}[r]|{\mathsize{\Huge}{\actsonright}} & \calA^w_n 
    \ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Compatibility with the Action on $F_n$} \label{par:action}
Let $\glos{\FA_n}$ denote the (degree-completed) free, associative (but
not commutative) algebra on the generators $\glos{x_1,\dots,x_n}$. Then
there is an ``expansion'' $Z\colon F_n\to \FA_n$ defined by $\xi_i\mapsto
e^{x_i}$ (see~\cite{Lin:Expansions} and the related ``Magnus Expansion''
of~\cite{MagnusKarrasSolitar:CGT}). Also, there is a right action\footnote{In the language of
\cite{Bar-NatanDancso:WKO2}, we will say that $\FA_n=\gr F_n$ and
that when the actions involved are regarded as instances of the algebraic
structure ``one monoid acting on another'', we have that \linebreak
$\left(\FA_n\actsonright\calA^w_n\right)=\grs\left(F_n\actsonright
\wB_n\right)$.} of
$\calA^w_n$ on $\FA_n$ defined on generators by $x_i\tau=x_{\tau i}$
for $\tau\in S_n$ and by $x_ja_{ij}=[x_i,x_j]$ and $x_ka_{ij}=0$ for
$k\neq j$ and extended by the Leibniz rule to the rest of $\FA_n$ and
then multiplicatively to the rest of $\calA^w_n$.

\begin{exercise}  Use the definition of the action in
\eqref{eq:ConceptualPsi} and the commutative diagrams of paragraphs
\ref{par:theta}, \ref{par:Delta} and~\ref{par:iota} to show that the
diagram of paragraph~\ref{par:action} is also commutative.
\end{exercise}

\pagebreak[2]

\parpic[r]{$\begin{array}{c}
  \pstex{StrandDoubling} \\
  \xymatrix{
    \wB_n \ar[r]^<>(0.5){u_k} \ar[d]_Z   & \wB_{n+1} \ar[d]^Z    \\
    \calA^w_n \ar[r]_<>(0.5){u_k}        & \calA^w_{n+1}
      \ar@{}[ul]|{\text{\huge$\not\circlearrowleft$}}
  }
\end{array}$}
\paragraph{Unzipping a Strand} \label{par:unzip} Given $k$ between $1$ and
$n$, let $\glos{u_k}\colon \wB_n\to\wB_{n+1}$ be the operation of ``unzipping
the $k$th strand'', briefly defined on the right\footnote{Unzipping
a knotted zipper turns a single band into two parallel ones. This
operation is also known as ``strand doubling'', but for compatibility with
operations that will be introduced later, we prefer ``unzipping''.}. The
induced operation $u_k\colon \calA^w_n\to\calA^w_{n+1}$ is also shown on
the right --- if an arrow starts (or ends) on the strand being doubled,
it is replaced by a sum of two arrows that start (or end) on either
of the two ``daughter strands''. This is performed for each arrow
independently; so if there are $t$ arrows touching the $k$th strands in
a diagram $D$, then $u_kD$ will be a sum of $2^t$ diagrams.

In some sense, much of this current series of papers as well as
the works of Kashiwara and Vergne~\cite{KashiwaraVergne:Conjecture}
and Alekseev and Torossian~\cite{AlekseevTorossian:KashiwaraVergne}
are about coming to grips with the fact that $Z$ is {\bf not}
compatible with $u_k$ (that the diagram on the right is {\bf not}
commutative). Indeed, let $x:=a_{13}$ and $y:=a_{23}$ be as on the
right, and let $s$ be the permutation $21$ and $\tau$ the permutation
$231$.  Then $d_1Z(\overcrossing)=d_1(e^{a_{12}}s)=e^{x+y}\tau$
while $Z(d_1\overcrossing)=e^ye^x\tau$. So the failure of $d_1$
and $Z$ to commute is the ill-behaviour of the exponential function
when its arguments do not commute, which is measured by the
BCH formula, central to both~\cite{KashiwaraVergne:Conjecture}
and~\cite{AlekseevTorossian:KashiwaraVergne}.

\subsubsection{Power and Injectivity}
The following theorem is due to Berceanu and
Papadima~\cite[Theorem~5.4]{BerceanuPapadima:BraidPermutation}; a variant
of this theorem is also true for ordinary braids~\cite{Kohno:deRham,
Bar-Natan:Homotopy, HabeggerMasbaum:Milnor}, and can be proven by
similar means:

\begin{theorem} $Z\colon \wB_n\to\calA^w_n$ is injective. In other words, finite
type invariants separate w-braids.
\end{theorem}

\begin{proof}The statement follows immediately from the faithfulness of
the action $F_n\actsonright\wB_n$, from the compatibility
of $Z$ with this action, and from the injectivity
of $Z\colon F_n\to\FA_n$ (the latter is well known, see
e.g.~\cite[Theorem~5.6]{MagnusKarrasSolitar:CGT}\footnote{Though notice
that we use $\xi_i\mapsto e^{x_i}$ whereas
\cite[Theorem~5.6]{MagnusKarrasSolitar:CGT} uses $\xi_i\mapsto 1+x_i$. The
\cite{MagnusKarrasSolitar:CGT} injectivity proof holds in our case
just as well.} and~\cite{Lin:Expansions}). Indeed, if $B_1$ and $B_2$
are w-braids and $Z(B_1)=Z(B_2)$, then $Z(\xi)Z(B_1)=Z(\xi)Z(B_2)$ for
any $\xi\in F_n$. Therefore $\forall\xi\, Z(\xi\sslash B_1)=Z(\xi\sslash
B_2)$, therefore $\forall\xi\,\xi\sslash B_1=\xi\sslash B_2$, therefore
$B_1=B_2$. \qed
\end{proof}

\begin{remark} Apart from the easy fact that $\calA^w_n$ can be computed
degree by degree in exponential time, we do not know a simple formula for
the dimension of the degree $m$ piece of $\calA^w_n$ or a natural basis of
that space. This compares unfavourably with the situation for ordinary
braids (see e.g.~\cite{Bar-Natan:Braids}). Also compare with
Problem~\ref{prob:wCombing} and with Remark~\ref{rem:GutierrezKrstic}.
\end{remark}

\subsubsection{Uniqueness} There is certainly not a unique expansion for
w-braids --- if $Z_1$ is an expansion and $P$ is any degree-increasing
linear map $\calA^w\to\calA^w$ (a ``pollution'' map), then $Z_2:=(I+P)\circ
Z_1$ is also an expansion, where $I\colon \calA^w\to\calA^w$ is the
identity. But that's all, and if we require a bit more, even that
freedom disappears.

\begin{proposition} If $Z_{1,2}\colon \wB_n\to\calA^w_n$ are expansions then
there exists a degree-increasing linear map $P\colon \calA^w\to\calA^w$ such
that $Z_2=(I+P)\circ Z_1$.
\end{proposition}

\begin{proof} (Sketch).  Let $\widehat{\wB_n}$ be the unipotent completion
of $\wB_n$. That is, let $\bbQ\wB_n$ be the algebra of formal linear
combinations of w-braids, let $\calI$ be the ideal in $\bbQ\wB_n$
generated by $\semivirtualover=\overcrossing-\virtualcrossing$ and by
$\semivirtualunder=\virtualcrossing-\undercrossing$, and set
\[ \widehat{\wB_n}:=
  \underleftarrow{\lim}_{m\to\infty} \bbQ\wB_n \left/\calI^m\right..
\]
Here $\widehat{\wB_n}$ is filtered with
$\calF_m\widehat{\wB_n}:=\underleftarrow{\lim}_{m'>m} \calI^m
\left/\calI^{m'}\right..$ An ``expansion'' can be re-interpreted as an
``isomorphism of $\widehat{\wB_n}$ and $\calA^w_n$ as filtered vector
spaces''. Always, any two isomorphisms differ by an automorphism of the
target space, and that's the essence of $I+P$. \qed
\end{proof}

\begin{proposition} If $Z_{1,2}\colon \wB_n\to\calA^w_n$ are homomorphic
expansions that commute with braid cloning (Paragraph~\ref{par:Delta}) and
with strand insertion (Paragraph~\ref{par:iota}), then \linebreak $Z_1=Z_2$.
\end{proposition}

\begin{proof} (Sketch). A homomorphic expansion that commutes with strand
insertions is determined by its values on the generators $\overcrossing$,
$\undercrossing$ and $\virtualcrossing$ of $\wB_2$. Commutativity
with braid cloning (see Paragraph \ref{par:Delta}) implies that these values must be, up to permuting
the strands, group like: that is, the exponentials of primitives. But
the only primitives are $a_{12}$ and $a_{21}$, and one may easily
verify that there is only one way to arrange these so that $Z$
will respect $\virtualcrossing^2=\overcrossing\cdot\undercrossing=1$ and
$\semivirtualover\mapsto\rightarrowdiagram+(\text{higher degree terms})$. \qed
\end{proof}

\subsubsection{The Group of Non-Horizontal Flying Rings}
\label{subsubsec:NonHorRings}
Let
$\glos{Y_n}$ denote the space of all placements of $n$ numbered disjoint
oriented unlinked geometric circles in $\bbR^3$. Such a placement
is determined by the centres in $\bbR^3$ of the circles, the radii,
and a unit normal vector for each circle pointing in the positive
direction, so $\dim Y_n=3n+n+3n=7n$.  $S_n \ltimes \bbZ_2^n$ acts on
$Y_n$ by permuting the circles and mapping each circle to its image in
either an orientation-preserving or an orientation-reversing way. Let
$\glos{\tilde{Y}_n}$ denote the quotient $Y_n/S_n \ltimes \bbZ_2^n$.
The fundamental group $\pi_1(\tilde{Y}_n)$ can be thought of as the
``group of flippable flying rings''.  Without loss of generality, we
can assume that the basepoint is chosen to be a horizontal placement.
We want to study the relationship of this group to $\wB_n$.

\begin{theorem} The group
$\pi_1(\tilde{Y}_n)$ is a $\bbZ_2^n$-extension of $\wB_n$, generated
by $s_i$, $\sigma_{i}$ (for $1\leq i \leq n-1)$, and $\glos{w_i}$ (``flips''),
for $1\leq i \leq n$; with the relations as above, and in addition:
\[
 w_i^2=1; \qquad w_iw_j=w_jw_i; \qquad w_js_i=s_iw_j \quad \text{when } \quad i\neq j, j+1;
\]
\[
 w_is_i=s_iw_{i+1}; \qquad w_{i+1}s_i=s_iw_i; \quad w_j\sigma_{i}=\sigma_{i}w_j \quad \text{if } \quad j \neq i, i+1;
\]
\[
w_{i+1}\sigma_{i}=\sigma_{i}w_{i}; \quad \text{yet } \quad
w_i\sigma_{i}=s_i\sigma_i^{-1}s_iw_{i+1}.
\]
\end{theorem}
The two most interesting flip relations in pictures:
\begin{equation}\label{eq:FlipRels}
  \raisebox{-10mm}{\input figs/FlipRels.pstex_t}
\end{equation}

\parpic[r]{\input{figs/FlippingRing.pstex_t}}
Instead of a proof, we provide some heuristics.
Since each circle starts out in a horizontal position and returns 
to a horizontal position, there is an integer number of 
``flips'' they do in between, these are the generators $w_i$, as 
shown on the right.

The first relation says that a double flip is homotopic to doing nothing.
Technically, there are two different directions of flips, and they are the 
same via this (non-obvious but true) relation. The rest of the first line is 
obvious: flips of different rings commute, and if
two rings fly around each other while another one flips, the order of these
events can be switched by homotopy. The second line says that, if two rings trade
places with no interaction while one flips, then the order of these events can be 
switched as well. However, we have to re-number the flip to conform to the
strand labelling convention.

The only subtle point is how flips interact with crossings. First of all,
if one ring flies through another while a third one flips, the order clearly
does not matter. If a ring flies through another and also flips, the 
order can be switched. However, if ring $A$ flips and then ring $B$ flies 
through it, this is homotopic to ring $B$ flying through ring $A$
from the other direction and then ring $A$ flipping. In other words, commuting
$\sigma_i$ with $w_i$ changes the ``sign of the crossing'' in the sense of
Exercise \ref{ex:swBn}. This gives the last, and the only truly non-commutative flip 
relation.

\parpic[r]{\input{figs/Wen.pstex_t}}
To explain why the flip is denoted by $w$, let us consider the alternative
description by ribbon tubes. A flipping ring traces a so called 
wen\footnote{The term wen was coined by Kanenobu and Shima in 
\cite{KanenobuShima:TwoFiltrationsR2K}}
in $\bbR^4$. A wen is a Klein bottle cut along a meridian circle, 
as shown. The wen is embedded in $\bbR^4$.

Finally, note that $\pi_1Y_n$ is exactly the pure $w$-braid group
$\PwB_n$: since each ring has to return to its original position and
orientation, each does an even number of flips.  The flips (or wens)
can all be moved to the bottoms of the braid diagram strands (to the
bottoms of the tubes, to the beginning of words), at a possible cost, as
specified by Equation~\eqref{eq:FlipRels}. Once together, they pairwise
cancel each other.  As a result, this group can be thought of as not
containing wens at all.

\subsubsection{The Relationship with u-Braids} \label{subsubsec:RelWithu} 
For
the sake of ignoring strand permutations, we restrict our
attention to pure braids.

\parpic[r]{$\xymatrix{
  \PuB \ar@{.>}[r]^{Z^u} \ar[d]^a & \calA^u \ar[d]^\alpha \\
  \PwB \ar[r]^{Z^w} & \calA^w
}$}
By Section \ref{subsubsec:FTAlgebraic}, for any expansion $Z^u\colon
\PuB_n \to \calA^u_n$ (where $\PuB_n$ is the ``usual'' braid group
and $\calA^u_n$ is the algebra of horizontal chord diagrams on $n$
strands), there is a square of maps as shown on the right. Here $Z^w$
is the expansion constructed in Section~\ref{subsec:wBraidExpansion},
the left vertical map $\glos{a}$ is the composition of the inclusion
and projection maps $\PuB_n \to \PvB_n \to \PwB_n$. The map $\glos{\alpha}$
is the induced map by the functoriality of $\grs$, as noted
after Exercise \ref{ex:BraidsAlgApproach}.  The reader can verify that
$\alpha$ maps each chord to the sum of its two possible directed versions.

Note that this square is {\it not} commutative for any choice of $Z^u$ even
in degree 2: the image of a crossing under $Z^w$ is outside the image 
of $\alpha$.

\parpic[r]{\input{figs/uwsquare2.pstex_t}}
More specifically, for any choice $c$ of a ``parenthesization'' of $n$
points, the KZ-construction / Kontsevich integral (see for example
\cite{Bar-Natan:NAT}) returns an expansion $Z_c^u$ of $u$-braids. We shall 
see in \cite{Bar-NatanDancso:WKO2} (Proposition 4.15 there) 
that for any choice of $c$, the two
compositions $\alpha \circ Z_c^u$ and $Z^w \circ a$ are ``conjugate in a
bigger space'': there is a map $i$ from $\calA^w$ to a larger space of
``non-horizontal arrow diagrams'', and in this space the images of the
above composites are conjugate.  However, we are not certain that $i$
is an injection, and whether the conjugation leaves the $i$-image of
$\calA^w$ invariant, and so we do not know if the two compositions differ
merely by an outer automorphism of $\calA^w$.