\draftcut
\section{Introduction} \label{sec:intro}
This is the second in a series of papers on w-knotted objects. In the
first paper \cite{Bar-NatanDancso:WKO1}, we took a classical approach
to studying finite type invariants of w-braids and w-knots and proved
that the universal finite type invariant for w-knots is essentially the
Alexander polynomial. In this paper we will study finite type invariants
of w-tangles and w-tangled foams from a more algebraic point of view,
and prove that ``homomorphic'' universal finite type invariants of
w-tangled foams are in one-to-one correspondence with solutions to
the (Alekseev-Torossian version of) the Kashiwara-Vergne problem in
Lie theory. Mathematically, this paper does not depend on the results
of \cite{Bar-NatanDancso:WKO1} in any significant way, and the reader
familiar with the theory of finite type invariants will have no difficulty
reading this paper without having read \cite{Bar-NatanDancso:WKO1}.
However, since this paper starts with an abstract re-phrasing of the
well-known finite type story in terms of general algebraic structures,
readers who need an introduction to finite type invariants may find
it more pleasant to read \cite{Bar-NatanDancso:WKO1} first (especially
Sections \ref{1-sec:intro}, \ref{1-sec:w-braids} and
\ref{1-subsec:VirtualKnots}--\ref{1-subsec:LieAlgebras}).
\subsection{Motivation and hopes}
This article and its siblings \cite{Bar-NatanDancso:WKO1} and \cite{Bar-NatanDancso:WKO3} are
efforts towards a larger goal. Namely, we believe 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~\cite{Drinfeld:QuantumGroups},
and 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.
This work was inspired by Alekseev and Torossian's results \cite{AlekseevTorossian:KashiwaraVergne} on
Drinfel'd associators and the Kashiwara-Vergne conjecture, both of which fall into the aforementioned
class of ``difficult equations in graded spaces''. The Kashiwara-Vergne conjecture ---
proposed in 1978 \cite{KashiwaraVergne:Conjecture} and proven in 2006 by Alekseev and Meinrenken \cite{AlekseevMeinrenken:KV} ---
has strong implications in Lie theory and harmonic analysis, and is a cousin of the Duflo isomorphism, which was shown to be
knot-theoretic in~\cite{Bar-NatanLeThurston:TwoApplications}.
We also know that Drinfel'd's theory
of associators~\cite{Drinfeld:QuasiHopf} 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},
or of knotted trivalent graphs~\cite{Dancso:KIforKTG}.
In Section \ref{sec:w-foams} we will re-interpret the Kashiwara-Vergne
conjecture as the problem of finding a ``homomorphic'' universal finite type invariant of
a class of w-knotted trivalent graphs (more accurately named w-tangled foams).
This result fits into a bigger picture incorporating usual, virtual and w-knotted objects and
their theories of finite type invariants, connected by the inclusion map from usual
to virtual, and the projection from virtual to w-knotted objects. In a sense that will be made precise in
Section \ref{sec:generalities}, usual and w-knotted objects with this mapping form
a unified algebraic structure, and the relationship between Drinfel'd associators
and the Kashiwara-Vergne conjecture is explained as a theory of finite type invariants for
this larger structure. This will be the topic of Section \ref{subsec:KTG}.
We are optimistic that this paper is a step towards re-interpreting the
work of Etingof and Kazhdan~\cite{EtingofKazhdan:BialgebrasI}
on quantization of Lie bi-algebras as a
construction of a well-behaved universal finite type invariant of
virtual knots~\cite{Kauffman:VirtualKnotTheory, Kuperberg:VirtualLink} or of a similar class
of virtually knotted objects.
However, w-knotted objects are quite interesting in their own right,
both topologically and algebraically:
they are related to combinatorial group
theory, 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{BrendleHatcher:RingsAndWickets, FennRimanyiRourke:BraidPermutation,
Goldsmith:MotionGroups, McCool:BasisConjugating, Satoh:RibbonTorusKnots}.
In \cite{Bar-NatanDancso:WKO1} we studied the universal finite type invariants of w-braids and w-knots,
the latter of which turns out to be essentially the Alexander polynomial.
A more thorough introduction about our ``hopes and dreams'' and the u-v-w big picture can also be
found in \cite{Bar-NatanDancso:WKO1}.
\subsection{A brief overview and large-scale explanation}
We are going to start by developing the algebraic ingredients of the paper in Section \ref{sec:generalities}.
The general notion of an {\em algebraic structure} lets us treat spaces of a topological or diagrammatic nature
in a unified algebraic manner. All of braids, w-braids, w-knots, w-tangles, etc., and their
associated chord- or arrow-diagrammatic counterparts form algebraic structures, and so do any number of these spaces combined, with
maps between them.
We then introduce {\em associated graded structures} with respect to a specific filtration, the machine which in our case takes an algebraic structure of
``topological nature'' (say, braids with $n$ strands) and produces the corresponding
diagrammatic space (for braids, horizontal chord diagrams on $n$ vertical strands). This is done by taking the associated graded space
with respect to a given filtration, namely the powers of the augmentation ideal in the algebraic structure.
An {\em expansion}, sometimes called a universal finite type invariant, is a map from an algebraic structure (in this case one of topological nature)
to its associated graded (a structure of combinatorial/diagrammatic nature), with a certain universality property.
A {\em homomorphic expansion} is one that is in addition
``well behaved'' with respect to the {\em operations} of the algebraic structure (such as composition and strand doubling for braids, for example).
The three main results of the paper are as follows:
\begin{enumerate}
\item
As mentioned before, our goal is to provide a topological framework for the Kashiwara-Vergne (KV) problem.
One of our main results is Theorem~\ref{thm:ATEquivalence}, in which we establish a bijection between certain homomorphic expansions
of {\em w-tangled foams} (introduced in Section \ref{sec:w-foams}) and solutions of the Kashiwara-Vergne equations. More precisely, ``certain''
homomorphic expansions means ones that are group-like (a commonly used condition), and subject to another minor technical condition.
Section \ref{sec:w-tangles} leads up to this result by studying the simpler case of {\em w-tangles} and identifying building blocks
of its associated graded structure as the spaces which appear in the \cite{AlekseevTorossian:KashiwaraVergne} formulation of the KV equations.
\item
In Theorem \ref{thm:WenATEquivalence} we study an unoriented version of w-tangled foams, and prove that homomorphic expansions
for this space (group-like and subject to the some minor conditions) are in one-to-one correspondence with solutions to the KV problem
with {\em even Duflo function}. This sets the stage for perhaps the most interesting result of the paper:
\item Section \ref{subsec:wTFcompatibility} marries the theory above with the theory of ordinary (not w-) knotted trivalent graphs (KTGs).
For technical reasons explained in Section \ref{sec:w-foams}, we work with a signed version of KTGs (sKTG).
Roughly speaking, homomorphic expansions for sKTGs are determined by a {\em Drinfel'd associator}. Furthermore, sKTGs map naturally
into w-tangled foams.
\noindent
In Theorem \ref{thm:ZuwCompatible} we prove that any homomorphic expansion of sKTGs coming from a {\em horizontal chord} associator
has a compatible homomorphic expansion of w-tangled foams, and furthermore, these expansions are in one-to-one correspondence
with {\em symmetric} solutions of the KV problem. This gives a topological explanation for the relationship between
Drinfel'd associators and the KV conjecture.
\end{enumerate}
We note that in \cite{Bar-NatanDancso:WKO3}
we'll further capitalize on these insights to provide a topological proof and interpretation for
Alekseev, Enriquez and Torossian's explicit solutions
for the KV conjecture in terms of associators \cite{AlekseevEnriquezTorossian:ExplicitSolutions}.
Several of the structures of a topological nature
in this paper (w-tangles and w-foams) are introduced as {\em Reidemeister theories}. That is, the spaces are built from pictorial
generators (such as crossings) which can be connected arbitrarily, and the resulting pictures are then factored out by certain relations
(``Reidemeister moves''). Technically speaking, this is done using the framework of {\em circuit algebras} (similar to planar algebras but
without the planarity requirement) which are introduced in Section \ref{sec:generalities}.
One of the fundamental theorems of classical knot theory is Reidemeister's theorem, which states that isotopy classes of knots are
in bijection with {\em knot diagrams} modulo Reidemeister moves. In our case, w-knotted objects have a Reidemeister description and
a topological interpretation in terms of ribbon knotted tubes in $\bbR^4$. However, the analogue of the Reidemeister theorem, i.e. the
statement that these
two interpretations coincide, is only known for w-braids \cite{McCool:BasisConjugating, Dahm:GeneralBraid, BrendleHatcher:RingsAndWickets}.
For w-tangles and w-foams (and w-knots as well) there is a map $\delta$ from the Reidemeister presentation to the appropriate class of ribbon
2-knotted objects in $\bbR^4$. In our case this means that all the generators
have a local topological interpretation and the relations represent isotopies. The map $\delta$ is certainly a surjection,
but it is only conjectured to be injective (in other words, it is possible that some relations are missing).
The main difficulty in proving the injectivity of $\delta$ lies
in the management of the ribbon structure. A ribbon 2-knot is a knotted sphere or long tube in $\bbR^4$ which admits a filling with only certain
types of singularities. While there are Reidemeister theorems for general 2-knots in $\bbR^4$ \cite{CarterSaito:KnottedSurfaces},
the techniques don't translate well to ribbon 2-knots, mainly because it is not well understood how different ribbon structures (fillings) of
the same ribbon 2-knot can be obtained from each other through Reidemeister type moves. The completion of such a theorem would be of great interest.
We suspect that even if $\delta$ is not injective, the present set of generators and relations describes a set of ribbon-knotted
tubes in $\bbR^4$ with possibly some extra combinatorial information, similarly to how, say, dropping the $R1$ relation in classical knot theory
results in a Reidemeister theory for framed knots with rotation numbers.
The paper is organized as follows: we start with a discussion of general algebraic structures,
associated graded structures, expansions (universal finite type invariants) and ``circuit algebras'' in Section \ref{sec:generalities}.
In Section \ref{sec:w-tangles} we study w-tangles and identify some of the spaces \cite{AlekseevTorossian:KashiwaraVergne}
where the KV conjecture ``lives'' as the spaces of ``arrow diagrams'' (the w-analogue of chord diagrams)
for certain w-tangles. In Section \ref{sec:w-foams} we study w-tangled foams and we prove the main theorems discussed above.
For more detailed information
consult the ``Section Summary'' paragraphs at the beginning
of each of the sections. A glossary of notation is on
page~\pageref{sec:glossary}.
\subsection{Corrigendum}\label{subsec:Corr} Yusuke Kuno and Haruko Miyazawa pointed out an error in the topological interpretation of w-tangled foams. Following several in-depth conversations with them, we made improvements to Section~\ref{sec:w-foams}, and submitted a Corrigendum \cite{Bar-NatanDancso:WKO2Corr}. In short, the ``adjoint'' $A_e$ operations of the published version of this paper were not well-defined with respect to the Reidemeister 4 relation. The simplest fix is to omit surface (two-dimensional) orientations and the adjoint operations entirely. The adjoint operations were used to relate the two ``vertex'' generators of w-foams, however, this can be achieved just as well by composing one of the vertices by ``wens'' (cut open Klein bottles). In this version of the paper we incorporated these changes into Section~\ref{sec:w-foams} for easier reading, and made several other stylistic improvements to the exposition.
\def\summaryalg{In this section we introduce the associated graded structure
of an ``arbitrary algebraic structure'' with respect to powers of its augmentation ideal
(Sections~\ref{subsec:AlgebraicStructures} and \ref{subsec:Grad})
and introduce the notions of ``expansions'' and ``homomorphic expansions'' (\ref{subsec:Expansions}).
Everything is so general that practically anything is an example,
yet our main goal is to set the language for the examples of w-tangles
and w-tangled foams, which appear later in this paper. Both of
these examples are types of ``circuit algebras'', and hence we
end this section with a general discussion of circuit algebras
(Sec.~\ref{subsec:CircuitAlgebras}).}
\def\summarytangles{In Sec.~\ref{subsec:vw-tangles} we introduce
v-tangles and w-tangles, the v- and w- counterparts of the
standard knot-theoretic notion of ``tangles'', and briefly discuss their
finite type invariants and their associated spaces of ``arrow diagrams'',
$\calA^v(\uparrow_n)$ and $\calA^w(\uparrow_n)$. We then construct a
homomorphic expansion $Z$, or a ``well-behaved'' universal finite type
invariant for w-tangles. The only algebraic tool we need to
use is $\exp(a):=\sum a^n/n!$ (Sec.~\ref{subsec:vw-tangles}
is in fact a routine extension of parts of
\cite[Section~\ref{1-sec:w-knots}]{Bar-NatanDancso:WKO1}).
In Sec.~\ref{subsec:ATSpaces} we show that
$\calA^w(\uparrow_n)\cong\calU(\fraka_n\oplus\tder_n\ltimes\attr_n)$,
where $\fraka_n$ is an Abelian algebra of rank $n$ and where
$\tder_n$ and $\attr_n$, two of the primary spaces used by Alekseev
and Torossian~\cite{AlekseevTorossian:KashiwaraVergne}, have simple
descriptions in terms of cyclic words and free Lie algebras. We also show
that some functionals studied in~\cite{AlekseevTorossian:KashiwaraVergne},
$\divop$ and $j$, have natural interpretations in our language.
In Section~\ref{subsec:sder} we discuss a subclass of w-tangles called ``special''
w-tangles, and relate them by similar means to Alekseev and Torossian's
$\sder_n$ and to ``tree level'' ordinary Vassiliev theory. A local topological interpretation is described in Sec.~\ref{subsec:TangleTopology} and the
uniqueness of $Z$ is studied in Sec.~\ref{subsec:UniquenessForTangles}.}
\def\summaryfoams{In this section we add ``foam vertices'' to w-tangles (and a few lesser
things as well) and ask the same questions we asked before;
primarily, ``is there a homomorphic expansion?''. As we shall
see, in the current context this question is equivalent to the
Alekseev-Torossian~\cite{AlekseevTorossian:KashiwaraVergne} version
of the Kashiwara-Vergne~\cite{KashiwaraVergne:Conjecture} problem and
explains the relationship between these topics and Drinfel'd's theory
of associators. This section was corrected and improved post-publication: see Section~\ref{subsec:Corr} and \cite{Bar-NatanDancso:WKO2Corr}.}
\subsection{Acknowledgement} We thank Yusuke Kuno and Haruko Miyazawa for their significant contribution which helped us improve Section 4 post-publication. We wish to thank Anton Alekseev, Jana
Archibald, Scott Carter, Karene Chu, Iva Halacheva, Joel Kamnitzer,
Lou Kauffman, Peter Lee, Louis Leung, Jean-Baptiste Meilhan, Dylan Thurston, Lucy Zhang
and the anonymous referees for comments and suggestions.