\draftcut
\section{Introduction} \label{sec:intro}
\subsection{Dreams} \label{subsec:dreams} We
have a dream\footnote{Understanding
the authors' history and psychology ought never be necessary to
understand their papers, yet it may be helpful. Nothing material
in the rest of this paper relies on Section~\ref{subsec:dreams}.},
at least partially founded on reality, that many of the difficult
algebraic equations in mathematics, especially those that are
written in graded spaces, more especially those that are related in
one way or another to quantum groups,
and even more especially those related to the work of Etingof and
Kazhdan~\cite{EtingofKazhdan:BialgebrasI}, can be understood, and indeed,
would appear more natural, in terms of finite type invariants of various
topological objects.
We believe this is the case for Drinfel'd's theory
of associators~\cite{Drinfeld:QuasiHopf}, which can be
interpreted as a theory of well-behaved universal finite type
invariants of parenthesized tangles\footnote{``$q$-tangles''
in~\cite{LeMurakami:Universal}, ``non-associative tangles''
in~\cite{Bar-Natan:NAT}.}~\cite{LeMurakami:Universal, Bar-Natan:NAT},
and as a theory of universal finite type invariants
of knotted trivalent graphs~\cite{Dancso:KIforKTG}.
We believe this is the case for Drinfel'd's ``Grothendieck-Teichm\"uller
group''~\cite{Drinfeld:GalQQ}, which is better understood as a
group of automorphisms of a certain algebraic structure, also
related to universal finite type invariants of parenthesized
tangles~\cite{Bar-Natan:Associators}.
And we're optimistic, indeed we believe, that sooner or later the
work of Etingof and Kazhdan~\cite{EtingofKazhdan:BialgebrasI}
on quantization of Lie bialgebras will be re-interpreted as a
construction of a well-behaved universal finite type invariant of virtual
knots~\cite{Kauffman:VirtualKnotTheory, ManturovIlyutko:VirtualKnotsBook}
or of some other class of virtually knotted objects. Some steps in that
direction were taken by Haviv~\cite{Haviv:DiagrammaticAnalogue}.
We have another dream, to construct a useful ``Algebraic Knot Theory''. As
at least a partial writeup exists~\cite{Bar-Natan:AKT-CFA},
we'll only state that an important ingredient necessary to
fulfil that dream would be a ``closed form''\footnote{The
phrase ``closed form'' in itself requires an explanation. See
Section~\ref{subsec:ClosedForm}. \label{foot:ClosedForm}} formula for an
associator, at least in some reduced sense. Formulae for associators or
reduced associators were in themselves the goal of several studies
undertaken for various other reasons~\cite{LeMurakami:HOMFLY,
Lieberum:gl11, Kurlin:CompressedAssociators, LeeP:ClosedForm}.
\draftcut \subsection{Stories}
Thus, the first named author, DBN, was absolutely delighted
when in January 2008 Anton Alekseev described to him his joint
work~\cite{AlekseevTorossian:KashiwaraVergne} with Charles
Torossian --- Anton told DBN that they found a relationship between the
Kashiwara-Vergne conjecture~\cite{KashiwaraVergne:Conjecture},
a cousin of the Duflo isomorphism (which DBN already knew to be
knot-theoretic~\cite{Bar-NatanLeThurston:TwoApplications}), and
associators taking values in a space called $\sder$, which he could
identify as ``tree-level Jacobi diagrams'', also a knot-theoretic
space related to the Milnor invariants~\cite{Bar-Natan:Homotopy,
HabeggerMasbaum:Milnor}. What's more, Anton told DBN that in certain
quotient spaces the Kashiwara-Vergne conjecture can be solved explicitly;
this should lead to some explicit associators!
So DBN spent the following several months trying to
understand~\cite{AlekseevTorossian:KashiwaraVergne} which eventually led to this sequence of papers.
One main thing we learned is that
the Alekseev-Torossian paper, and with it the Kashiwara-Vergne (KV)
conjecture, fit very nicely with our first dream recalled above,
about interpreting algebra in terms of knot theory. Indeed much
of~\cite{AlekseevTorossian:KashiwaraVergne} can be reformulated as a
construction and a discussion of a well-behaved universal finite type
invariant\footnote{The notation $Z$ for universal finite type invarants
comes from the famous universal finite type invariant of classical links, the Kontsevich itegral.} $Z$
of a certain class of knotted objects (which we will call
w-knotted), a certain natural quotient of the space of virtual knots
(more precisely, virtual trivalent tangles): this will be the subject of the second
paper in the series. It is also possible to provide a topological interpretation
(and independent topological proof) of the \cite{AlekseevEnriquezTorossian:ExplicitSolutions}
formula for explicit solutions to the KV problem in terms of associators. This will be done in
the third paper.
And our hopes remain high
that later we (or somebody else) will be able to exploit this relationship
in directions compatible with our second dream recalled above, on the
construction of an ``algebraic knot theory''.
The story, in fact, is prettier than we were hoping for, for it has the
following additional qualities:
\begin{myitemize}
\item w-Knotted objects are quite interesting in themselves: as
stated in the abstract, they are related to combinatorial group
theory via ``basis-conjugating'' automorphisms of a free group $F_n$,
to groups of movies of flying rings in $\bbR^3$, and more generally, to
certain classes of knotted surfaces in $\bbR^4$. The references include
\cite{Goldsmith:MotionGroups, McCool:BasisConjugating, FennRimanyiRourke:BraidPermutation,
Satoh:RibbonTorusKnots, BrendleHatcher:RingsAndWickets}.
\item The ``chord diagrams'' for w-knotted objects (really, these are ``arrow
diagrams'') describe formulae for invariant tensors in spaces pertaining to
not-necessarily-metrized Lie algebras in much the same way as ordinary
chord diagrams for ordinary knotted objects describe formulae for invariant
tensors in spaces pertaining to metrized Lie algebras. This observation is
bound to have further implications.
\item Arrow diagrams also describe the Feynman diagrams of topological BF
theory \cite{CattaneoCotta-RamusinoMartellini:Alexander, CCFM:BF34} and of a
certain class of Chern-Simons theories~\cite{Naot:BF}. Thus, it is likely that
our story is directly related to quantum field theory\footnote{Some
non-perturbative relations between BF theory and w-knots was discussed by
Baez, Wise and Crans~\cite{BaezWiseCrans:ExoticStatistics}.}.
\item The main objective of this paper is to prove that, when composed
with the map from knots to w-knots, $Z$ becomes the
Alexander polynomial. For links, it becomes an invariant stronger than the
multi-variable Alexander polynomial which contains the multi-variable
Alexander polynomial as an easily identifiable reduction.
\item On other
w-knotted objects $Z$ has easily identifiable reductions that can be
considered as ``Alexander polynomials'' with good behaviour relative
to various knot-theoretic operations --- cablings, compositions
of tangles, etc. There is also a certain specific reduction of $Z$
that can be considered as an ``ultimate Alexander polynomial'' ---
in the appropriate sense, it is the minimal extension of the Alexander
polynomial to other knotted objects which is well behaved under a whole
slew of knot theoretic operations, including the ones named above. See
\cite{Bar-NatanSelmani:MetaMonoids, Bar-Natan:KBH}.
\ifpub{\item The true value of w-knots, though, is likely to emerge
later, for we expect them to serve as a \underline{w}armup example
for what we expect will be even more interesting --- the study of
\underline{v}irtual knots, or v-knots. We expect v-knotted objects
to provide the global context whose associated graded structure will
be the Etingof-Kazhdan theory of deformation quantization of Lie
bialgebras~\cite{EtingofKazhdan:BialgebrasI}.}{}
\end{myitemize}
\begin{figure}
\[
\def\uT{\parbox[t]{1.875in}{\small
Ordinary (\underline{u}sual) knotted objects in 3D --- braids,
knots, links, tangles, knotted graphs, etc.
}}
\def\vT{\parbox[t]{1.875in}{\small
\underline{V}irtual knotted objects --- ``algebraic'' knotted objects,
or ``not specifically embedded'' knotted objects; knots drawn on a
surface, modulo stabilization.
}}
\def\wT{\parbox[t]{1.875in}{\small
Ribbon knotted objects in 4D; ``flying rings''. Like v, but also with
``overcrossings commute''.
}}
\def\uC{\parbox[t]{1.875in}{\small
Chord diagrams and Jacobi diagrams, modulo $4T$, $STU$, $IHX$, etc.
}}
\def\vC{\parbox[t]{1.875in}{\small
Arrow diagrams and v-Jacobi diagrams, modulo $6T$ and various
``directed'' $STU$s and $IHX$s, etc.
}}
\def\wC{\parbox[t]{1.875in}{\small
Like v, but also with ``tails commute''. Only ``two in one out''
internal vertices.
}}
\def\uL{\parbox[t]{1.875in}{\small
Finite dimensional metrized Lie algebras, representations, and
associated spaces.
}}
\def\vL{\parbox[t]{1.875in}{\small
Finite dimensional Lie bialgebras, representations, and associated
spaces.
}}
\def\wL{\parbox[t]{1.875in}{\small
Finite dimensional co-commutative Lie bialgebras (i.e.,
$\frakg\ltimes\frakg^\ast$), representations, and associated
spaces.
}}
\def\uH{\parbox[t]{1.875in}{\small
The Drinfel'd theory of associators.
}}
\def\vH{\parbox[t]{1.875in}{\small
Likely, quantum groups and the Etingof-Kazhdan theory of quantization
of Lie bialgebras.
}}
\def\wH{\parbox[t]{1.875in}{\small
The Kashiwara-Vergne-Alekseev-Torossian theory of convolutions on Lie
groups and Lie algebras.
}}
\pstex{uvw}
\]
\caption{The u-v-w Stories} \label{fig:uvw}
\end{figure}
\draftcut \subsection{The Bigger Picture}
Parallel to the w-story run the possibly more significant u-story
and v-story. The u-story is about u-knots, or more generally,
u-knotted objects (braids, links, tangles, etc.), where ``u'' stands
for \underline{u}sual; hence the u-story is about classical knot
theory. The v-story is about v-knots, or more generally, v-knotted
objects, where ``v'' stands for \underline{v}irtual, in the sense of
Kauffman~\cite{Kauffman:VirtualKnotTheory} (see
also~\cite{ManturovIlyutko:VirtualKnotsBook}).
The stories of u-, v-, and w-knotted objects, are quite different from
each other. Yet they can be told along similar lines --- first the knots
(topology), then their finite type invariants and their ``chord diagrams''
(combinatorics), then those map into certain universal enveloping
algebras and similar spaces associated with various classes of Lie
algebras (low algebra), and finally, in order to construct a ``good''
universal finite type invariant, in each case one has to confront a
certain deeper algebraic subject (high algebra). These stories are
summarized in a table form in Figure~\ref{fig:uvw}.
u-Knots map into v-knots, and v-knots map into w-knots\footnote{Though
the composition ``$u\to v\to w$'' is not $0$. In fact, the composed
map $u\to w$ is injective. u-Knots, for example, are determined by the
fundamental groups of their complements plus ``peripheral systems''
(or alternatively, by their ``quandles''~\cite{Joyce:TheKnotQuandle}),
and this information is easily recovered from the w-knot images of
u-knots. Similar considerations apply to other classes of u-knotted
objects.}. The other parts of our stories, the ``combinatorics'' and
``low algebra'' and ``high algebra'' rows of Figure~\ref{fig:uvw}, are
likewise related, and this relationship is a crucial part of our overall
theme. Thus, we cannot and will not tell the w-story in isolation, and
while it is central to this article, we will necessarily also include
some episodes from the u and v series.
\subsection{Plans} In this paper we study w-braids and w-knots; the main
result is Theorem \ref{thm:Alexander}, which states that the universal
finite type invariant of w-knots is essentially the Alexander polynomial.
However, starting with braids and taking a classical approach to
finite type invariants, this paper also serves as a gentle introduction
to the subsequent papers and in particular to \cite{Bar-NatanDancso:WKO2}
where we will present a more algebraic point of view.
For more detailed information on the content
consult the ``Section Summary'' paragraphs below and at the beginning
of each section. An ``odds and ends'' section and a glossary of notation
follows the main sections.
\def\summarybraids{This section is largely a compilation of existing
literature, though we also introduce the language of arrow diagrams that
we use throughout the rest of the paper. In Sections~\ref{subsec:VirtualBraids}
and~\ref{subsec:wBraids} we define v-braids and then w-braids and
survey their relationship with basis-conjugating automorphisms of free
groups and with ``the group of (horizontal) flying rings in $\bbR^3$''
(really, a group of knotted tubes in $\bbR^4$). In Section ~\ref{subsec:FT4Braids}
we play the usual game of introducing finite type invariants, weight
systems, chord diagrams (arrow diagrams, for this case), and 4T-like
relations. In Section~\ref{subsec:wBraidExpansion} we define and construct
a universal finite type invariant $Z$ for w-braids --- it turns out
that the only algebraic tool we need to use is the formal exponential
function $\exp(a):=\sum a^n/n!$. In Section~\ref{subsec:bcomments} we study
some good algebraic properties of $Z$, its injectivity, and its
uniqueness, and we conclude with the slight modifications needed for the
study of non-horizontal flying rings.}
\def\summaryknots{In Section~\ref{subsec:VirtualKnots} we define v-knots and
w-knots (long v-knots and long w-knots, to be precise) and discuss a map
$v\to w$. In Section~\ref{subsec:FTforvwKnots} we determine the space of ``chord
diagrams'' for w-knots to be the space $\calA^w(\uparrow)$ of arrow
diagrams modulo $\aft$ and TC relations and in Section~\ref{subsec:SomeDimensions}
we compute some relevant dimensions. In Section~\ref{subsec:Jacobi} we show
that $\calA^w(\uparrow)$ can be re-interpreted as a space of trivalent
graphs modulo STU- and IHX-like relations, and is therefore related
to Lie algebras (Section~\ref{subsec:LieAlgebras}). This allows us to
completely determine $\calA^w(\uparrow)$. With no difficulty
in Section~\ref{subsec:Z4Knots} we construct a universal finite type
invariant for w-knots. With a bit of further difficulty we show in
Section~\ref{subsec:Alexander} that it is essentially equal to the Alexander
polynomial.}
%\noindent
%{\small \begin{multicols}{2}
{\bf Section~\ref{sec:w-braids}, w-Braids.} %(page~\pageref{sec:w-braids})
\summarybraids
{\bf Section~\ref{sec:w-knots}, w-Knots.} %(page~\pageref{sec:w-knots})
\summaryknots
%$\end{multicols}}
\subsection{Acknowledgements} We wish to thank the Anonymous Referee, the
Copy Editor, Anton Alekseev, Jana Archibald, Scott Carter, Karene Chu, Iva
Halacheva, Joel Kamnitzer, Lou Kauffman, Peter Lee, Louis Leung, Vassily
Olegovich Manturov, Jean-Baptiste Meilhan, Sherilyn Tamagawa, Dylan
Thurston, Daniel Tubbenhauer, Huan Vo, and Lucy Zhang for comments and suggestions.