\def\proj{{\operatorname{proj}\,}}
\def\projs{{\operatorname{proj}}}
\section*{Recycling}
\begin{exercise} Do the same for the obviously-defined ``w-links'',
excluding the material about the Alexander polynomial. Note that the wheels
that are obtained in the case of w-links have legs coloured by the
components of the w-link in question. Hence if there is more than one
component, the number of such wheels grows exponentially in the degree and
thus $Z$ contains more information than can be coded in a polynomial of
even a multi-variable polynomial.
\end{exercise}
\vskip -5mm
\parpic[r]{\raisebox{-14mm}{$\pstex{CC}$}}
\begin{conjecture} In the case of ordinary links seen as w-links,
if we mod out the target space of $Z$ by the ``Commutators Commute''
relation shown on the right, what remains of the wheels part of $Z$
is precisely the multi-variable Alexander polynomial.
\end{conjecture}
Note that $D \in \tder_n$ is never an arrow on a single strand (these are
elements of $\fraka_n$),
and hence $\operatorname{div}D$ is never a 1-wheel, more precisely it never
has a degree 1 component. Thus, even
though the target space of div is $\attr_n/(\text{deg }1)$, we can just as
well think of it as a
map to $\attr_n$ itself.
\subsection{The Injectivity of $i_u\colon F_n\to\wB_{n+1}$}
\label{subsec:FreeInW}
Just for completeness, we sketch here an algebraic proof of the
injectivity of the map $i_u\colon F_n\to\wB_{n+1}$ discussed in
Section~\ref{subsubsec:McCool}. There's some circularity in our argument
--- we need this injectivity in order to motivate the definition of the
map $\Psi\colon \wB_n\to\Aut(F_n)$, and in the proof below we use $\Psi$
to prove the injectivity of $i_u$. But $\Psi$ exists regardless of how
its definition is motivated, and it can be shown to be well defined by
explicitly verifying that it respects the relations defining $\wB_n$. So
our proof is logically valid.
\begin{claim} The map $i_u\colon F_n\to\wB_{n+1}$ is injective.
\end{claim}
\begin{proof} (sketch). Let $H$ be the subgroup of $\wB_{n+1}$ MORE
\end{proof}
\subsection{Finite Type Invariants of v-Braids and w-Braids, in some
Detail} \label{subsec:FTDetails}
As mentioned in Section~\ref{subsec:wBraids}, w-braids are v-braids modulo
an additional relation. So we start with a discussion of finite type
invariants of v-braids. For simplicity we take our base ring to be $\bbQ$;
everywhere we could replace it by an arbitrary field of characteristic
$0$\footnote{Or using the variation of constants method, we can simply
declare that $\bbQ$ is an arbitrary field of characteristic $0$.},
and many definitions make sense also over $\bbZ$ or even with $\bbQ$
replaced by an arbitrary Abelian group.
\subsubsection{Basic Definitions.}
Let $\bbQ\vB_n$ denote group ring of $\vB_n$, the algebra of formal linear
combinations of elements of $\vB_n$, and let $\bbQ S_n$ be the group
ring of $S_n$. The skeleton homomorphism of Remark~\ref{rem:Skeleton}
extends to a homomorphism $\varsigma\colon \bbQ\vB_n\to\bbQ S_n$. Let $\calI$
(or $\calI_n$ when we need to be more precise) denote the kernel of
the skeleton homomorphism; it is the ideal in $\bbQ\vB_n$ generated
by formal differences of v-braids having the same skeleton. One may
easily check that $\calI$ is generated by differences of the form
$\overcrossing-\virtualcrossing$ and $\virtualcrossing-\undercrossing$.
Following~\cite{GoussarovPolyakViro:VirtualKnots} we call such differences
``semi-virtual crossings'' and denote them by $\semivirtualover$
and $\semivirtualunder$, respectively\footnote{The signs in
$\semivirtualover\leftrightarrow\overcrossing-\virtualcrossing$ and
$\semivirtualunder\leftrightarrow\virtualcrossing-\undercrossing$ are
``crossings come with their sign and their virtual counterparts come with
the opposite sign''.}. In a similar manner, for any natural number $m$
the $m$th power $\calI^m$ of $\calI$ is generated by ``$m$-fold iterated
differences'' of v-braids, or equally well, by ``$m$-singular v-braids'',
which are v-braids that also have exactly $m$ semi-virtual crossings
(subject to relations which we don't need to specify).
Let $V\colon \vB_n\to A$ be an invariant of v-braids with values in some vector
space $A$. We say that $V$ is ``of type $m$'' (for some $m\in\bbZ_{\geq
0}$) if its linear extension to $\bbQ\vB_n$ vanishes on $\calI^{m+1}$
(alternatively, on all $(m+1)$-singular v-braids, in clear analogy with the
standard definition of finite type invariants). If $V$ is of type $m$
for some unspecified $m$, we say that $V$ is ``of finite type''. Given
a type $m$ invariant $V$, we can restrict it to $\calI^m$ and as it
vanishes on $\calI^{m+1}$, this restriction can be regarded as an element
of $\left(\calI^m/\calI^{m+1}\right)^\star$. If two type $m$ invariants
define the same element of $\left(\calI^m/\calI^{m+1}\right)^\star$ then
their difference vanishes on $\calI^m$, and so it is an invariant of type
$m-1$. Thus it is clear that an understanding of $\calI^m/\calI^{m+1}$
will be instrumental to an inductive understanding of finite type
invariants. Hence the following definition.
\begin{definition} The projectivization\footnote{Why ``projectivization''?
See Section~\ref{subsec:Projectivization}.} $\proj\vB_n$ is the direct sum
\[ \proj\vB_n:=\bigoplus_{m\geq 0} \calI^m/\calI^{m+1}. \]
Note that throughout this paper, whenever we write an infinite direct sum,
we automatically complete it. Therefore an element in $\proj\vB_n$ is an
infinite sequence $D=(D_0, D_1,\dots)$, where $D_m\in\calI^m/\calI^{m+1}$.
The projectivization $\proj\vB_n$ is a graded space, with the degree $m$
piece being $\calI^m/\calI^{m+1}$.
\end{definition}
We proceed with the study of $\proj\vB_n$ (and thus of finite type
invariants of v-braids) in three steps. In
Section~\ref{subsubsec:ArrowDiagrams} we introduce a space $\calD^v_n$ and
a surjection $\rho_0\colon \calD^v_n\to\proj\vB_n$. In Section~\ref{subsubsec:6T}
we find some relations in $\ker\rho_0$, most notably the $6T$ relation, and
introduce the quotient $\calA^v_n:=\calD^v_n/6T$. And then in
Section~\ref{subsubsec:UFTI} we introduce the notion of a ``universal
finite type invariant'' and explain how the existence of such a gadget
proves that $\proj\vB_n$ is isomorphic to $\calA^v_n$ (in a more
traditional language this is the statement that every weight system
integrates to an invariant).
Unfortunately, we do not know if there is a universal finite type invariant
of v-braids. Thus in Section~\ref{subsec:wbraids} we return to the subject
of w-braids and prove the weaker statement that there exists a universal
finite type invariant of w-braids.
\subsubsection{Arrow Diagrams.} \label{subsubsec:ArrowDiagrams}
We are looking for a space that will surject on $\calI^m/\calI^{m+1}$. In
other words, we are looking for a set of generators for $\calI^m$, and
we are willing to call two such generators the same if their difference
is in $\calI^{m+1}$. But that's easy. Left and right multiples of the
formal differences $\semivirtualover=\overcrossing-\virtualcrossing$
and $\semivirtualunder=\virtualcrossing-\undercrossing$
generate $I$, so products of the schematic form
\begin{equation} \label{eq:GeneratingProduct}
B_0 (\semivirtualover|\semivirtualunder) B_1
(\semivirtualover|\semivirtualunder) B_2 \cdots
B_{m-1} (\semivirtualover|\semivirtualunder) B_m
\end{equation}
\parpic[r]{\input figs/SemiVirtRels.pstex_t }
\noindent generate $\calI^m$ (here $(\semivirtualover|\semivirtualunder)$
means ``either a $\semivirtualover$ or a $\semivirtualunder$'', and there
are exactly $m$ of those in any product). Furthermore, inside such a
product any $B_k$ can be replaced by any other v-braid $B'_k$ having the
same skeleton (e.g., with $\varsigma(B_k)$), for then $B_k-B'_k\in\calI$
and the whole product changes by something in $\calI^{m+1}$. Also,
the relations in~\eqref{eq:R3} and in~\eqref{eq:MixedRelations} imply
the relations shown on the right for $\semivirtualover$, and similar
relations for $\semivirtualunder$. With this freedom, a product as
in~\eqref{eq:GeneratingProduct} is determined by the strand-placements
of the $\semivirtualover$~s and the $\semivirtualunder$ s. That is,
for each semi-virtual crossing in such a product, we only need to
know which strand number is the ``over'' strand, which strand number
is the ``under'' strand, and a sign that determines whether it is the
positive semi-virtual $\semivirtualover$ or the negative semi-virtual
$\semivirtualunder$. This motivates the following definition.
\begin{definition} A ``horizontal $m$-arrow diagrams'' (analogues to the
``chord diagrams'' of, say, \cite{Bar-Natan:OnVassiliev}) is an ordered
pair $(D,\beta)$ in which $D$ is a word of length $m$ in the alphabet
$\{a^+_{ij},a^-_{ij}\colon i,j\in\{1,\ldots,n\},\,i\neq j\}$ and $\beta$ is a
permutation in $S_n$. Let $\calD_m^{vh}$ be the space of formal linear
combinations of horizontal $m$-arrow diagrams. We usually use a pictorial
notation for horizontal arrow diagram, as demonstrated in
Figure~\ref{fig:Dvh}.
\end{definition}
\begin{figure}
\parpic[r]{\hspace{-5mm}\raisebox{-29mm}{$\pstex{Dvh}$}}
\caption{
The horizontal $3$-arrow diagram
$(D,\beta)=$ $(a^+_{12}a^-_{41}a^+_{23},\,3421)$ and its image via
$\rho_0$. The first arrow, $a^+_{12}$ starts at strand $1$, ends
at strand $2$ and carries a $+$ sign, so it is mapped to a positive
semi-virtual crossing of strand $1$ over strand $2$. Likewise the second
arrow $a^-_{41}$ maps to a negative semi-virtual crossing of strand
$4$ over strand $1$, and $a^+_{23}$ to a positive semi-virtual
crossing of strand $2$ over strand $3$. We also show one possible
choice for a representative of the image of $\rho_0(D,\beta)$ in
$\calI^m/\calI^{m+1}$: it is a v-braid with semi-virtual crossings as
specified by $D$ and whose overall skeleton is $3421$.
} \label{fig:Dvh}
\end{figure}
There is a surjection $\rho_0\colon \calD_m^{vh}\to\calI^m/\calI^{m+1}$. The
definition of $\rho_0$ is suggested by the first paragraph of this section
and an example is shown in Figure~\ref{fig:Dvh}; we will skip the formal
definition here. We also skip the formal proof of the surjectivity of
$\rho_0$.
Finally, consider the product $\semivirtualover\cdot\semivirtualunder$ and
use the second Reidemeister move for both virtual and non-virtual
crossings:
\[
\semivirtualover\semivirtualunder
= (\overcrossing-\virtualcrossing)(\virtualcrossing-\undercrossing)
= \overcrossing\virtualcrossing+\virtualcrossing\undercrossing
- \overcrossing\undercrossing - \virtualcrossing\virtualcrossing
= (\overcrossing\virtualcrossing-1) + (\virtualcrossing\undercrossing)
= \semivirtualover\virtualcrossing - \virtualcrossing\semivirtualunder.
\]
If a total of $m-1$ further semi-virtual crossings are multiplied into this
equality on the left and on the right, along with arbitrary further
crossings and virtual crossings, the left hand side of the equality becomes
a member of $\calI^{m+1}$, and therefore, as a member of
$\calI^m/\calI^{m+1}$, it is $0$. Thus with ``$\ldots$'' standing for
extras added on the left and on the right, we have that in
$\calI^m/\calI^{m+1}$,
\[ 0 =
\ldots(
\semivirtualover\virtualcrossing-\virtualcrossing\semivirtualunder
)\ldots
= \rho_0(\ldots??\ldots)
\]
MORE.
\subsubsection{The $6T$ Relations.} \label{subsubsec:6T}
MORE.
\subsubsection{The Notion of a Universal Finite Type Invariant.}
\label{subsubsec:UFTI}
MORE.
\subsection{Finite type invariants of w-braids} \label{subsec:wbraids}
MORE.