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\begin{document}
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\title{{Finite Type Invariants
of w-Knotted Objects II: the Double Tree Construction}}
\author{Dror~Bar-Natan and Zsuzsanna Dancso}
\address{
Department of Mathematics\\
University of Toronto\\
Toronto Ontario M5S 2E4\\
Canada
}
\email{drorbn@math.toronto.edu, zsuzsi@math.toronto.edu}
\urladdr{http://www.math.toronto.edu/~drorbn, http://www.math.toronto.edu/zsuzsi}
\date{first edition in future, this edition Jan.~27,~2014. The
\arXiv{????.????} edition may be older}
\subjclass{57M25}
\keywords{
virtual knots,
w-braids,
w-knots,
w-tangles,
knotted graphs,
finite type invariants,
Kashiwara-Vergne,
associators,
double tree,
free Lie algebras%
}
\thanks{This work was partially supported by NSERC grant RGPIN 262178.
Electronic version and related files at~\cite{DoubleTree},
\url{http://www.math.toronto.edu/~drorbn/papers/DoubleTree/}.
}
\begin{abstract}
\input abstract.tex
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\subsection{Topology}
We begin by describing a chain of maps from ``parenthesized braids'' to ``(signed) knotted trivalent graphs'' to ``w-tangled foams'':
$$\calK:=\{\uPB \stackrel{\cl}{\longrightarrow} \sKTG \stackrel{a}{\longrightarrow} \wTFe\}.$$
Let us first briefly elaborate on each of these spaces and maps.
Parenthesized braids are braids are braids whose ends are ordered along two lines, the ``bottom and the ''top``, along with parenthetizations
of the endpoints on the bottom and on the top. Two examples are shown in Figure \ref{fig:PBexample}. Parentehesized braids form a category whose
objects are parenthetizations, morphisms are the parenthesized braids themselves, and composition is given by stacking. In addition to stacking,
there are several operations defined on parenthesized braids: strand addition, removal and doubling. A detailed introduction
to parenthesized braids is in \cite{Bar-Natan:GT1}.
\begin{figure}
\input{figs/PBexample.pstex_t}
\caption{Two examples of parenthesized braids. Note that by convention the parenthetization can be read from the distance scales between the endpoints
of the braid, and so we are going to omit the parentheses in the future.}\label{fig:PBexample}
\end{figure}
\parpic[r]{\input{figs/KTGExample.pstex_t}}
Trivalent graphs are oriented graphs with three edges meeting at each vertex and whose vertices are equipped with a cyclic orientation of the edges.
A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph into $\bbR^3$.
KTGs are studied from a finite type invariant point of view in \cite{Bar-NatanDancso:KTG}. In this paper we use a version of KTGs that was
introduced and studied in Section 6.6 of \wko, namely trivalent $(1,1)$-tangles with some extra combinatorial information:
signs assigned to the trivalent vertices. We call this space $\sKTG$, as in \wko. An example is shown on the right.
The space $\sKTG$ is also equipped with several operations: tangle insertion, edge unzip, and edge orientation switch.
The space $\wTFe$ is a minor extension of $\wTFo$ studied in Section 6 of \wko, and will be introduced in detail in Section
\ref{sec:wTFe}. It can be described as a planar algebra generated by certain features (various kinds of crossings and vertices, as well as ``caps'')
modulo certain relations (``Reidemeister moves'') and equipped with a number of auxiliary operations beyond planar algebra composition. This Reidemeister
theory conjecturally represents knotted tubes in $\bbR^4$ with singular ``foam vertices'', caps, and attached one-dimensional strings.
The map $\cl: \uPB \to \sKTG$ is the ``closure map''. Given a parenthesized braid, close up its top and bottom each by a tree according to the parentetization;
this produces a $\sKTG$ with the convention that all strands are oriented upwards, bottom vertices are negative, and top vertices are positive.
An example is shown below.
\begin{center}
\input{figs/Closure.pstex_t}
\end{center}
The map $a: \sKTG \to \wTFe$ arises combinatorially from the fact that all $\sKTG$ diagrams can be interpreted as elements of $\wTFe$, and all
$sKTG$ Reidemeister moves remain true in $\wTFe$. Topologically, it is an extended version of Satoh's tubing map, described in Remark 3.1.1 of
\wko.
\subsection{Algebra}
The chain of maps $\calK$ is an example of a general ``algebraic structure'', as defined in Section 4.1 of \wko. An algebraic
structure consists of a collection of objects belonging to a number of ``spaces'' or ``different kinds'', and operations that may be unary, binary,
multinary or zeronary, between these spaces. In this case there are many spaces (or kinds of objects): for example, parenthesized braids with specified
bottom and top parenthetizations form one space, so do knottings of a given trvalent graph. There is also a large collection of operations, consisting
of all the internal operations of $\uPB$, $\sKTG$ and $\wTFe$, as well as the maps $a$ and $\cl$.
In Section 4.2 and 4.3 of \wko we discuss projectivizations and expansions for general algebraic structures. A projectivization
is the associated graded space taken with respect to the filtration by powers of the ``augmentation ideal''. For the spaces $\uPB$, $\sKTG$ and $\wTFe$, the
projectivizations $\calA^{hor}$, $\calA^{u}$ and $\calA^{sw}$ are the spaces of ``horizontal chord diagrams on parenthesized strands'', ``chord diagrams
on trivalent skeleta'', and ``arrow diagrams'', as described in \cite{Bar-Natan:GT1}, Section 6.6 of \wko, and Section \ref{sec:wTFe}
of this paper, respectively. As a result, the projectivization of $\calK$ is the structure
$$\calA:=\{\calA^{hor}\stackrel{\cl}{\longrightarrow} \calA^{u} \stackrel{\alpha}{\longrightarrow} \calA^{sw}\},$$
where $\cl$ and $\alpha$ are the maps induced by $\cl$ and $a$, respectively. More specifically, $\cl$ is the ``closure of chord diagrams'', and $\alpha$ is
``sending each chord to the sum of its two possible oriantations''.
An expansion is a filtration-respecting map from an algebraic structure (where linear combinations of objects of the same kind are allowed) to its
projectivization, satisfying a certain
non-degeneracy property. Expansions are also called universal finite type invariants in knot theory. A homomorphic extension also behaves well with respect to
the operations of the algebraic structure, that is, it intertwines each operation with its induced counterpart on the projectivization.
Hence, a homomorphic expansion $Z: \calK \to \calA$ is a triple of homomorphic expansions $Z^b, Z^u,$ and $Z^w$ for $\uPB$, $\sKTG$ and $\wTFe$, respectively, so that
the following diagram commutes:
\begin{equation}
\xymatrix{
\uPB \ar[r]^{\cl} \ar[d]^{Z^b}
& \sKTG \ar[r]^{a} \ar[d]^{Z^u}
& \wTFe \ar[d]^{Z^w} \\
\calA^{hor} \ar[r]^{\cl}
& \calA^u \ar[r]^{\alpha}
& \calA^w
}
\end{equation}
We recall (see \cite{Bar-Natan:GT1}) that a homomorphic expansion $Z^b$ for parenthesized braids is determined by a ``horizontal chord associator''
$\Phi=Z^b(\raisebox{-1mm}{\input{figs/Phi.pstex_t}})$.
A homomorphic expansion $Z^u$ of $\sKTG$ is also determined by an associator (horizontal chords or not; see Section 6.6 of \wko),
so the significance of left commutative square is to force $Z^u$ to come from a horizontal chord associator. In turn, $Z^w$ (roughly speaking) is determined by a solution
$V=Z^w(\raisebox{-1mm}{\input{figs/PlusVertex.pstex_t}})$ to the Kashiwara-Vergne problem (see Section 6 of \wko),
and the goal of this paper is to prove the following theorem:
\begin{theorem}\label{thm:main}
\begin{enumerate}
\item Assuming that $Z: \calK \to \calA$ exists, $Z$ is determined by $\Phi$. In other words, there is a formula for $V$ in terms of $\Phi$, assuming that $V$ exists.
\item Said formula is the formula proven in \cite{AlekseevEnriquezTorossian:ExplicitSolutions}.
\item Every $Z^b$ extends to a $Z$.
\end{enumerate}
\end{theorem}
The key to the proof of the theorem is to show that the generator \raisebox{-.7mm}{\input{figs/PlusVertex.pstex_t}} of $\wTFe$ can be expressed in terms of the
generator \raisebox{-.7mm}{\input{figs/Phi.pstex_t}} of $\uPB$ and the operations of $\calK$. Assuming that $Z$ exists, this yields a formula for $V$ in terms
of $\Phi$. The expression of \raisebox{-.7mm}{\input{figs/PlusVertex.pstex_t}} in terms of \raisebox{-.7mm}{\input{figs/Phi.pstex_t}} uses a
``Double Tree Construction'', which will be discussed in Section \ref{sec:DoubleTree}. For now, let us display a picture with no explanation:
\begin{center}
\input{figs/PhiToV.pstex_t}
\end{center}
\section{The spaces $\wTFe$ and $\calA^{sw}$ in more detail}\label{sec:wTFe}
As we mentioned in the introduction, $\wTFe$ is a minor extension of the space $\wTFo$ studied in Section 6 of \wko. It can be introduced
as a planar algebra or a circuit algebra, we will do the latter as it is simpler and more concise. Circuit algebras are defined in Section 4.4 of
\wko; in short, they are similar to a planar algebras but the ``connecting strands'' are allowed to cross. As in \wko,
each generator and relation of $\wTFe$ has a local topological interpretation. In \wko, $\wTFo$ represented knotted tubes in $\bbR^4$ with
``foam vertices'' and ``capped ends''. Two-dimensional tubes will be denoted by thick lines and one dimensional ones by thin red lines. The space $\wTFe$ extends
$\wTFo$ by adding one-dimensional strands to the picture. Note that one dimensional strands cannot be knotted in $\bbR^4$, however, they can be knotted
{\em with} two-dimensional tubes.
\[
\wTFe=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/wTFgens.pstex_t}}
\right|
\parbox{1.2in}{\centering relations as in Section~\ref{subsec:wrels}}
\right|
\parbox{1.2in}{\centering operations as in Section~\ref{subsec:wops}}
\right\rangle.
\]
\subsection{The generators of $\wTFe$}\label{subsec:wgens}
We begin by discussing the local topological meaning of each generator shown above.
\parpic[l]{\includegraphics[height=5cm]{figs/TheVertex.ps}}
The first five generators are as described in Section 6.1.1 of \wko. Recall that knotted (more precisely, braided) tubes
in $\bbR^4$ can equivalently be thought of as movies of flying rings in $\bbR^3$. The two crossings stand for movies where two rings trade places
while the one corresponding to the under strand flies through the one corresponding to the over strand. The dotted end represents a tube ``capped off''
by a disk. The fourth and fifth generators stand for singular ``foam vertices'', and will be referred to as the positive and negative vertex, respectively.
The positive vertex represents the movie shown on the left: the right ring approaches the left ring from below, flies inside it and merges with it. To be
completely precise, $\wTFe$ as a circuit algebra has more generators than shown above: the vertices appear with all possible orientations of the strands.
However, all other versions can be obtained from the ones shown above using ``orientation switch'' operations (to be discussed in Section \ref{subsec:wops}).
\parpic[r]{\input figs/MixedCrossings.pstex_t }
The red (thin) strands denote one dimensional strings in $\bbR^4$, or ``flying points in $\bbR^3$''. The crossings between the two types of strands
(sixth and seventh generators) denote ``points flying through rings''. They are both shown on the right in band notation (see section 5.4 of \wko
for an explanation of band notation). For example, the bottom left picture means ``the point on the approaches the ring on the left from below, flies
through the ring and out to the left above it.''
This explains why there are no generators with a thick strand crossing under a
thin red strand: a ring cannot fly through a point.
Next is a trivalent vertex of 1-dimensional strings in $\bbR^4$. Once again, this generator should be shown in all possible strand orientation combinations.
Finally, tha last generator is a ``mixed vertex'', in other words a one-dimesional string attached to the wall of a 2-dimensional tube.
\subsection{The relations}\label{subsec:wrels}
As a list, the relations for $\wTFe$ look the same as the relations for $\wTFo$: $\{$\Rs, R2, R3, R4, OC, CP$\}$. Recall that \Rs is the weak (framed) version
of the Reidemeister 1 move; R2 and R3 are the usual Reidemeister moves; R4 allowes moving a strand over or under a vertex. OC is the ``Overcrossings Commute''
relation, and CP (Cap Pullout) allows for pulling a capped strand out from under a crossing, as shown below:
\begin{center}
\input figs/CapRel.pstex_t
\end{center}
However, all relations should be interpreted in all
possible combinations of strand types, for example the lower strand of the Reidemeister two relation can either be thick black or thin red:
\begin{center}
\input figs/R2.pstex_t
\end{center}
Similarly, any of the bottom strands of the R3, R4, or OC relations may be thin red.
As in $\wTFo$, the relations all have local topological meaning and conjecturally $\wTFe$ is a Reidemeister theory for ribbon knotted tubes in $\bbR^4$
with caps, singular foam vertices and attached strings. For example, Reidemeister 2 with a thin red bottom strand is imposed because a point flying in through
a ring and then immediately flying back out is isotopic to not having any interaction between the point and ring at all.
\subsection{The operations}\label{subsec:wops}
Like $wTFo$, $wTFe$ is equipped with a set of auxiliary operations in addition to the circuit algebra structure.
The first of these is orientation reversal. For the thin (red) strands, this simply means reversing the direction of the strand.
For the thick (tube) strands, orientation switch comes in two flavours. Recall from \wko that in the topological interpretation of
$\wTFo$, each tube is oriented as a 2-dimensional surface, and also has a distinguished ``core'': a line along the tube which is oriented as
a 1-dimensional manifold and determines the ``direction'' or ``1-dimensional orientation'' of the tube. Both of these are determined by
the direction of the strand in the circuit algebra, via Satoh's tubing map.
Topologically, the operation ``orientation switch'', denoted
$S_e$ for a given strand $e$, acts by reversing both the (1-dimensional) direction and the (2-dimensional) orientation of the tube $e$.
Diagrammatically, this corresponds to simply reversing the direction of the corresponding strand $e$.
\parpic[r]{\input{figs/Adjoint.pstex_t}}
The ``adjoint'' operation, denoted $A_e$, on the other hand
only reverses the (1-dimensional) direction of the tube $e$, not the orientation as a surface. Diagrammatically, this manifests as reversing the strand
direction and adding two virtual crossings on either side of each crossing where $e$ crosses {\em over} another strand, as shown on the right (note
that the strand below $e$ may be thick or thin).
For more details on orientations and orientation switches, see \wko.
\parpic[r]{\input{figs/StringUnzip.pstex_t}}
The unzip operation $u_e$ doubles the strand $e$ using the
blackboard framing, and then attaches the ends of the doubled strand to the
connecting ones, as shown on the right. We restrict unzip to strands
whose two ending vertices are of different signs. (For the definition of
crossing and vertex signs, see Sections 5.4 and 6.1 of \wko.) Topologically,
the blackboard framing of the diagram induces a framing of the corresponding tube in $\bbR^4$
via Satoh's tubing map, and unzip is the act of ``pushing the tube off of itself slightly in the
framing direction''. Note that unzips preserve the ribbon property.
A related operation, {\it disk unzip}, is unzip done on a capped strand, pushing the tube off in the direction of the framing
(in diagram world, in the direction of the blackboard framing), as
before. An example in the line and band notations is shown below.
\begin{center}
\input{figs/CapUnzip.pstex_t}
\end{center}
We also allow the deletion of ``long linear'' strands, meaning strands that do not end in a vertex on either side.
So far all the operations we have introduced had already existed in $\wTFo$. The first new operation is called ``puncture'',
denoted $p_e$,
which diagrammatically simply turns the thick black strand $e$ into a thin red one. The corresponding topological picture
is ``puncturing a tube'', i.e., removing a small disk from it and retracting the rest to its core.
Any crossings where $e$ passes under another
strand are not affected, while crossings in which $e$ is the over strand turn into virtual crossings.
\parpic[r]{\input figs/PunctureRule.pstex_t }
For simplicity, we place a restriction on which strands can be punctured, namely at each (fully thick black) vertex puctures are only allowed for
one of the three meeting strands, as shown in the top row of the figure on the right. More general puctures could be allowed in a theory complete
with ``wens'', as in Section 6.5 of \wko. The bottom row of the same figure
shows what happens when puncturing one of the thick strands of a mixed vertex. Topologically, this is because the mixed
vertex represents a string attached to the ouside of a tube, so when puncturing $e$, the entire tube retracts to its core.
Finally, puncturing a capped tube makes it disappear.
\subsection{The projectivization $\calA^{sw}$}
As in \wko, the space $\wTFe$ is filtered by powers of the augmentation ideal and its associated graded space or projectivization,
denoted $\calA^{sw}$, is a ``space
of arrow diagrams on foam skeletons with strings''. As a circuit algebra, $\calA^{sw}$ is presented as follows:
\[
\wTFe=\CA\!\left.\left.\left\langle
\raisebox{-2mm}{\input{figs/AswGens.pstex_t}}
\right|
\parbox{0.8in}{\centering relations as below}
\right|
\parbox{0.9in}{\centering operations as below}
\right\rangle.
\]
The first and fifth generators are called single arrows and they are of degree one, while all others are ``skeleton features'' of degree zero.
The relations are almost the same as those for the projectivization of $\wTFo$: $\aft$ (the 4-Term relation), TC (Tails Commute), RI (Rotation
Invariance), CP (Cap Pullout), VI (Vertex Invariance), with the additional new relation TF (Tails Forbidden on strings).
The TC and $\aft$ relations are shown in Figure \ref{fig:TCand4T}, note that the 3rd strand in each term of the $\aft$ relation is ambiguous: it can be
either thick black or thin red, the relation applies in either case. VI is picturd in Figure \ref{fig:VI}: here the $\pm$ signs depend on the
strand orientations and the type of the
vertex and the types of each strand (thick black or thin red) is left ambiguous: the VI relation applies in all cases. Figure \ref{fig:RICPTF} shows
the other three relations: RI, CP and TF. Note that technically TF is not even a relation: there were no generators with an arrow tail on a thin red strand,
so saying that such an element vanishes is superfluous. However, without TF the VI relation would have to be stated for all the sub-cases of 0, 1 or 3 thin red
strands meeting at the vertex, instead of simply saying that arrow tails on these strands vanish. We prefer stating them this way as it is cleaner, even if it is a
slight abuse of notation.
\begin{figure}
\input{figs/TCand4T.pstex_t}
\caption{The TC and $\protect\aft$ relations}
\label{fig:TCand4T}
\end{figure}
\begin{figure}
\input{figs/VI.pstex_t}
\caption{The VI relation: the vertices and strands could be of any type, but the same throughout the relation.}
\label{fig:VI}
\end{figure}
\begin{figure}
\input{figs/RICPTF.pstex_t}
\caption{The RI relation, CP relation and the TF relation (which is not really a relation).}
\label{fig:RICPTF}
\end{figure}
As in \wko (see Definition 3.13), we define a ``w-Jacobi diagram'' (or just ``arrow diagram'') to be similar to by
also allowing tricalent chord vertices, each of which is equipped with a cyclic orientation. Denote the circuit algebra
of formal linear combinations of these w-Jacobi diagrams by, $\calA^{swt}$. Then, as in Theorem 6.5 in \wko, we have the
following bracket-rise theorem:
\begin{theorem}
The obvious inclusion of diagrams induces a circuit algebra isomorphism $\calA^{sw}\cong \calA ^{swt}$. Furthermore,
the $\aAS$ , $\aIHX$ and $\aSTU$ relations (see Figure \ref{fig:ASIHXSTU}) hold in $\calA^{swt}$.
\end{theorem}
\begin{figure}
\input{figs/ASIHXSTU.pstex_t}
\caption{The $\protect\aAS$, $\protect\aIHX$ and the two $\protect\aSTU$ rerations.}
\label{fig:ASIHXSTU}
\end{figure}
The proof is identical to the proof of Theorem 3.15 in \wko. In light of this isomorphism, we will drop the extra ``t'' from the
notation and use $\calA^{sw}$ to denote either of these spaces.
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