(243,279.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $N^{12}$;

(412,253) node [anchor=north west][inner sep=0.75pt]  [font=]  $N^{12}$;

(412,274) node [anchor=north west][inner sep=0.75pt]  [font=]  $C^{^{12}}$;

(156.82,222.92) node [anchor=north west][inner sep=0.75pt]    $=$;

(325.82,226.92) node [anchor=north west][inner sep=0.75pt]    $=$;

(67,132) node [anchor=north west][inner sep=0.75pt]    $G$;

(430,132) node [anchor=north west][inner sep=0.75pt]    $G$;

For any $G\in \Aut(\arrows)$ and $C\in \arrows(\uparrow)$, $N$ commutes with $C^{12}$.

It is also possible, and requires approximately the same effort, to deduce the equations of Proposition <ref> directly from the relations of $\arrows$. However, it would require significantly more work to show that this is a complete set of equations without relying on the [11] characterisation of expansions and Proposition <ref>.

Recall from Lemma <ref> (based on <cit.>) that there is an isomorphism of Hopf algebras
$$ \arrows(\uparrow_2)\stackrel{\Upsilon}\cong \widehat{U}(\cyc_2\rtimes(\tder_2\oplus \mathfrak{a}_2)),$$
where $\mathfrak{a}_2$ is central, and $\cyc_2$ is commutative.

For $N=G(\vertex)$, we write $\Upsilon(N)=\Upsilon (e^\eta)=:e^we^{n+a}$ for the image of $N$ in $\widehat{U}(\cyc_2\rtimes(\tder_2\oplus \mathfrak{a}_2)),$ where $w\in \cyc_2$, $n\in \tder_2$, and $a\in \mathfrak{a}_2$.

In Section <ref> we saw that the set of solutions to the Kashiwara–Vergne conjecture are in one-to-one correspondence with the $v$-small homomorphic expansions. To establish the $v$-small condition in the $\krv$ context, we make the following definition.

An automorphism $G\in \Aut(\arrows)$ is v-small if, for $N=G(\vertex)$, we have $\Upsilon(N)=e^we^n $. That is to say, $G\in\Aut(\A)$ is $v$-small if the projection of $\log \Upsilon(N)$ onto the central subalgebra $\mathfrak{a}_2$ is zero. Let $\Aut_v(\arrows)$ denote the subgroup of v-small elements of $\Aut(\arrows).$

The modified equation (<ref>) is identical to the equation (<ref>), but for $N$ in place of $V$.
The following result from <cit.> applies directly when one substitutes $N$ for $V$:

The equation (<ref>) is satisfied if and only if $\mathcal{J}(e^n)=e^{-2w}$.

The following is a straightforward implication of Proposition <ref> and <ref>:

For a circuit algebra automorphism $G: \arrows \to \arrows$, following are equivalent:

* $G\in \Aut_v(\arrows)$.

* For any v-small group-like homomorphic expansion $Z: \wf \to \arrows$, the composition $G\circ Z: \wf \rightarrow \arrows$ is also a v-small group-like homomorphic expansion.

* Denoting $G(\vertex)=N$ and $G(\upcap)=\Gamma$: the projection of $\log \Upsilon(N)$ to $\mathfrak{a}_2$ is zero, and $N$ and $\Gamma$ satisfy the equations (<ref>), (<ref>) and (<ref>) with $R=e^{\rightarrowdiagram}$.

We're now ready to state and prove the main theorem of this section:

There is an isomorphism of groups $\Aut_v(\arrows)\cong\krv$.

Let $G$ be an element in $\Aut_v(\arrows)$ with
\[G(\vertex)=N=e^\eta=\Upsilon^{-1}(e^we^n) \quad \text{and} \quad G(\upcap)=\Gamma=e^\mu=e^{\kappa^{-1}(\gamma)},\]
where we recall from Lemma <ref> that  $\gamma=\kappa(\mu) \in x^2\mathbb Q[[x]]$. Similarly, by Lemma <ref>, $e^\eta=\Upsilon^{-1}(e^we^n)$, with $n\in\tder_2$ and $w\in \cyc_2$. We define a map

\begin{equation}\label{eq:antiiso}\tag{$\overline{\Theta}$}\begin{tikzcd} \overline{\Theta}: \Aut_v(\arrows) \arrow[r] & \krv \end{tikzcd}\end{equation}
by setting $\overline{\Theta}(G)=(\alpha,s)\in \krv$ with  $\alpha=e^{n}$ and $s=2\gamma$. We will show that $\overline{\Theta}$ is an anti-isomorphism, and then compose with inversions to obtain and isomorphism.

The main difficulty is in showing that the map $\overline{\Theta}$ is well-defined, that is, that the pair $(\alpha,s)$ satisfies the defining equations of the group $\krv$. To do this, we use that the isomorphism $\Upsilon$ is compatible with the adjoint action of $\TAut_2$ on $\exp(\lie_2)$ in the following sense. Let $(\Upsilon^{-1})^{12}$ denote the composition
\[(\Upsilon^{-1})^{12}:\TAut_2 \xrightarrow{\Upsilon^{-1}} \arrows(\uparrow_2) \hookrightarrow \arrows(\uparrow_3),\]
where the inclusion $\arrows(\uparrow_2)\hookrightarrow\arrows(\uparrow_3)$ is the natural inclusion of arrow diagrams by adding an empty third strand.
As we saw in Remark <ref>, there is an inclusion $\log \iota: \lie_2 \hookrightarrow \arrows(\uparrow_3)$ defined on basis elements by $\log\iota(x)=a_{13}$, $\log\iota(y)=a_{23}$, where $a_{ij}$ denotes the horizontal arrow from strand $i$ to strand $j$, and the bracket in the Lie algebra of primitive elements of $\arrows(\uparrow_3)$ is the algebra commutator. The map $\iota: \exp(\lie_2) \to \arrows(\uparrow_3)$ is defined to be $e^{\log \iota}$.
With this notation, we recall from Remark <ref> that the following diagram commutes:

\begin{equation}\label{eq:TAutAction}
\begin{tikzcd}
\TAut_2 \times \exp(\lie_2)  \arrow[d, hookrightarrow, swap, "(\Upsilon^{-1})^{12}\times \iota"]\arrow[rr, "\ad"]  && \exp(\lie_2) \arrow[d, hookrightarrow, "\iota"]
\\
\arrows(\uparrow_3)\times \arrows(\uparrow_3) \arrow[rr, "\text{conj}"] && \arrows(\uparrow_3).
\end{tikzcd}
\end{equation}

Now, applying $\Upsilon$ to the (<ref>) equation, we obtain:

\begin{equation}\label{Upsilon R4'}e^we^ne^{x+y}e^{-n}e^{-w}=e^{x+y}.\end{equation}

Multiplying (<ref>) by $e^{-w}$ on the left and by $e^{w}$ on the right, and using that $[w,x]=[w,y]=0$, as $\cyc_2$ acts trivially on $\lie_2$, this simplifies to

\begin{equation*} e^ne^{x+y}e^{-n}=e^{x+y}. \end{equation*}

Therefore, $e^{x+y}=e^{-n}e^{x+y}e^n=\alpha(e^{x+y})$ by the definition of the action of $\alpha$.
Thus, we have shown $\alpha (e^{x+y})=e^{x+y}$: the first defining equation of the group $\krv$.

To verify that the pair $(\alpha, s)$ satisfies the second equation of the group $\krv$, recall from Lemma <ref> that there are linear isomorphisms $\kappa: \arrows(\upcap)
\xrightarrow{\cong} \cyc_1/\langle\xi\rangle$ and $\kappa: \arrows(\upcap_2)\xrightarrow{\cong}\cyc_2/\langle \xi, \zeta \rangle$, between arrow diagrams on capped strands and degree
completed cyclic words with no single-letter terms (see also Example 2.2 and 2.3 of [3]).  Thus, we will
write $\kappa (Z(\upcap))=\kappa (C)=e^{\trace(c(\xi))}$, and $\kappa (G(\upcap))=\kappa(\Gamma)=e^{\trace(\gamma(\xi))}$, where $c, \gamma\in \mathbb Q[[\xi]]$ have no linear terms.  It follows that $
\kappa (C^{12})=e^{\trace( c(\xi + \zeta))}$, and $\kappa(\Gamma^{12})=e^{\trace(\gamma(\xi + \zeta))}$, where $\trace:\widehat{\mathfrak{ass}}_2 \to \cyc_2$ is the linear trace operator.

Recall from Remark <ref> that $\arrows(\upcap_2)$ is a left $\arrows(\uparrow_2)$-module and the following diagram commutes:

\begin{equation}\label{eq:TAutOnWheels}
\begin{tikzcd}
\TAut_2 \times \exp(\cyc_2/\langle\xi,\zeta\rangle) \arrow[d, hookrightarrow, swap, "(\Upsilon^{-1})\times \kappa^{-1}"]\arrow[rr, "\cdot"]  && \exp(\cyc_2/\langle\xi,\zeta\rangle) \arrow[d, rightarrow, "\kappa^{-1}"]
\\
\arrows(\uparrow_2)\times \arrows(\upcap_2) \arrow[rr, "\text{stack}"] && \arrows(\upcap_2)
\end{tikzcd}
\end{equation}
where the top map is the basis conjugating action of $\TAut_2$ on $\exp(\cyc_2)$ as in Example<ref> and Remark <ref>.

Putting this all together, we see that applying $\kappa$ to the equation (<ref>) yields the following equation in $\cyc_2/\langle\xi,\zeta \rangle$:

\begin{equation}\label{eq:algebraC'}e^w e^n\cdot (e^{\trace(\gamma(\xi+\zeta))})=e^{\trace(\gamma(\xi))}e^{\trace(\gamma(\zeta))}.\end{equation}

Now, since the action of $\TAut_2$ on $\cyc_2$ comes from the action of $\TAut_2$ on $\exp(\lie_2)$ via taking traces, we have that $e^n\cdot e^{\xi+\zeta}=e^{\xi+\zeta}$ by the (<ref>) equation (and noticing that if $e^{-n}$ fixes $(\xi +\zeta)$ then so does $e^n$). Therefore,
$$e^n\cdot (e^{\trace(\gamma(\xi+\zeta))})=e^{\trace(\gamma(\xi+\zeta))},$$

and equation (<ref>) simplifies to

$$e^w =e^{\trace(\gamma(\xi))}e^{\trace(\gamma(\zeta))}e^{-\trace(\gamma(\xi+\zeta))}=e^{\trace(\gamma(\xi)+\gamma(\zeta)-\gamma(\xi+\zeta))}.$$

Proposition <ref> says that since $G$ respects the (<ref>) equation, we know that $\mathcal{J}(e^n)=e^{-2w}$ and thus $e^w=(\mathcal J(e^n))^{-1/2}=\mathcal J(\alpha)^{-1/2}$.

\begin{equation*}\log \mathcal J(\alpha)=J(\alpha)=\trace(2\gamma(\xi+\zeta)-2\gamma(\xi)-2\gamma(\zeta)),\end{equation*}

and therefore $s=2\gamma$ satisfies the second defining equation of $\krv$ with $\alpha=e^n$. This completes the proof that $\overline{\Theta}$ is well-defined.

It remains to show that the map $\overline{\Theta}$ is anti-multiplicative: given $G_1, G_2 \in \Aut_v(\arrows)$ with $G_1(\vertex)=N_1$ and $G_2(\vertex)=N_2$, then $(G_2\circ G_1)(\vertex)=N_1N_2$, where the multiplication is performed in $\arrows(\uparrow_2)$ (using Lemma <ref>). Similarly, if $G_1(\upcap)=e^{\kappa^{-1}(\gamma_1(x))}$ and $G_2(\upcap)=e^{\kappa^{-1}(\gamma_2(x))}$ then $(G_2\circ G_1)(\upcap)= e^{\kappa^{-1}((\gamma_1+\gamma_2)(x))}$. (We have omitted the trace notation in order to reduce clutter.) It follows that $\overline{ \Theta}: \Aut_v(\arrows) \to \krv $ is a group anti-homomorphism.

To show that $\overline{\Theta}$ is bijective, we will construct its inverse $\overline{\Theta}^{-1}: \krv \to \Aut_v(\arrows)$ as follows. Given a pair $(\alpha,s)\in \krv$, define an automorphism $G=\overline{\Theta}^{-1}(\alpha,s)\in \Aut_v(\arrows)$ by setting:

\[N=G(\vertex)=\Upsilon^{-1}\left((\mathcal J(\alpha))^{-1/2}\alpha\right) \quad \text{and}  \quad \Gamma=G(\upcap)=e^{\kappa^{-1}(s/2)}.\]

The map $\overline{\Theta}^{-1}$ is clearly inverse to $\overline{\Theta}$, and so it remains to show that it is well-defined.  By Proposition <ref> this amounts to verifying that $N$ and $\Gamma$ satisfy equations (<ref>), (<ref>) and (<ref>) and therefore define an automorphism of $\Aut_v(\arrows)$. Of these, (<ref>) is automatic by construction and by Proposition <ref>. Equation (<ref>) is satisfied by the commutativity of diagram (<ref>) and the first defining equation of $\krv$. Equation (<ref>) reduces to the second defining equation of $\krv$, using the commutativity of the diagram (<ref>).

To complete the proof and show we have a group isomorphism, we compose $\overline{\Theta}$ with the group inversion in $\krv$:

\begin{equation}\label{iso}\tag{$\Theta$}\begin{tikzcd} {\Theta}: \Aut_v(\arrows) \arrow[r, "\overline{\Theta}"] & \krv \arrow[r, "(-)^{-1}"] &\krv \end{tikzcd}\end{equation}

The map $\Theta$ is well-defined since $\krv$ is a group, and it is an isomorphism because it is a composite of two anti-isomorphisms. Explicitly, we have $\Theta(G)=(\alpha:=e^{-n},s:=-2\gamma)$, where $G$ is as in the beginning of the proof.

We show that the isomorphism from Theorem <ref> is compatible with the action of $\krv$ on $\text{SolKV}$. We briefly summarise <cit.> which identifies
the set $\text{SolKV}$ with the set of $v$-small group-like homomorphic expansions of $\wf$ as follows.

A $v$-small homomorphic expansion $Z$ evaluated at the vertex $\vertex$ determines a value $\Upsilon(Z(\vertex))=e^be^\nu$ with $b\in \cyc_2$ and $\nu\in \tder_2$, so $e^\nu \in \TAut_2$. The corresponding KV solution $F\in\text{SolKV}$ is given by $F_Z:=e^{-\nu}$.[The paper [11] follows the conventions of [3], where KV solutions are the inverses of of the elements defined to be in SolKV in [1]. Hence the negative in the exponent does not appear in [11].] Conversely, given an $F\in \text{SolKV}$, the corresponding homomorphic expansion $Z_F:\wf\to \arrows$ is defined by $Z_F(\vertex)=\Upsilon^{-1}((\mathcal J(F^{-1}))^{-1/2}F^{-1})$ and $Z_F(\upcap)$ is uniquely determined by $Z_F(\vertex)$.

Let $\mathcal{HE}$ denote the set of $v$-small, group-like homomorphic expansions $Z:\wf \to \arrows$ and let $\mathcal W$ denote the [11] identification $\mathcal{HE} \longrightarrow \text{SolKV}$ sending $Z\mapsto F_Z:=e^{-\nu}$.

The action of $\Aut_v(\arrows)$ on $\mathcal{HE}$ via post-composition, $Z\cdot G=G^{-1}\circ Z$, is compatible with the action of $\krv$ on $\textnormal{SolKV}$. In other words, the following square commutes:

\[
\begin{tikzcd}
\mathcal{HE} \times \Aut_v(\arrows)  \arrow[d, "\mathcal{W}\times \Theta", swap]  \arrow[r, "\cdot"]
& \mathcal{HE} \arrow[d, "\mathcal{W}"]
\\
\textnormal{SolKV} \times \krv  \arrow[r, "\cdot"] & \textnormal{SolKV}
\end{tikzcd}
\]

Let $(Z,G)\in \mathcal{HE} \times \Aut_v(\arrows)$, with $Z(\vertex)=V$, $G(\vertex)=N$, $\Upsilon(V)=e^be^\nu$, $\Upsilon(N)=e^we^n$. Then $\mathcal{W}(Z)=e^{-\nu}\in \text{SolKV}$, and $\Theta(G)=e^{-n}\in \krv$. Finally, the right action of KRV on SolKV is, as discussed above, $e^{-\nu}\cdot e^{-n}=e^ne^{-\nu}\in \text{SolKV}$.

On the other hand, $Z\cdot G=G^{-1}\circ Z$, and $G^{-1}\circ Z(\vertex)= VN^{-1}$. The latter statement uses the fact that $G(\rightarrowdiagram)=\rightarrowdiagram$, which in turn implies that $G^{-1}(\vertex)=N^{-1}$. Since $\Upsilon(VN^{-1})=e^be^\nu e^{-n}e^{-w}$, we have $\mathcal{W}(G^{-1}\circ Z)=(e^\nu e^{-n})^{-1}=e^n e^{-\nu}$. This agrees with the above, completing the proof.

§.§ The group $\kv$

In this section, we describe the symmetry group $\kv$ as a group of automorphisms of the completed circuit algebra of welded foams $\hatwf$. The structure of $\hatwf$ is considerably more complicated than its associated graded $\arrows$.  This makes direct computation of the  group of automorphisms of $\hatwf$ difficult. As such, we rely on Theorem <ref>, which states that homomorphic expansions $Z$ of $w$-foams induce circuit algebra isomorphisms $\widehat{Z}: \hatwf \to \arrows$.

Let $\Aut(\hatwf)$ denote the group of circuit algebra automorphisms $g:\hatwf\rightarrow \hatwf$ which preserve the set of group-like homomorphic expansions under pre-composition. Let $\Aut_v(\hatwf)$ denote the group of circuit algebra automorphisms of $\hatwf$ which preserve the set of group-like homomorphic expansions, and the $v$-small property, under pre-composition.

Note that the automorphisms in $\Aut(\hatwf)$ necessarily commute with the auxiliary operations on $\hatwf$ (unzips and orientation switches). The proof of this is identical to the proof of the corresponding statement in Proposition <ref>.

Any choice of homomorphic expansion $Z:\wf \to \arrows$ induces an isomorphism $\theta_Z: \Aut(\hatwf) \to \Aut(\arrows)$ given by $\theta_Z(g)=\widehat{Z}g\widehat{Z}^{-1}$, for $g\in \Aut(\hatwf)$. For a homomorphic expansion $Z$ satisfying the $v$-small condition, this restricts to an isomorphism $\theta_Z: \Aut_v(\hatwf)\to \Aut_v(\arrows)$.

This follows from the fact that $\widehat{Z}$ is a circuit algebra isomorphism which also intertwines the auxiliary operations (Proposition <ref>). Note that by [11] and [3], the existence of a KV solution implies that at least one homomorphic expansion exists.

Recall that the groups $\kv$ and $\krv$ act freely and transitively on $\text{SolKV}$ on the left and right respectively, as follows: for $a \in \kv$, $F \in \text{SolKV}$ and $\alpha \in \krv$, we have $a \cdot F:= F \circ a^{-1}$, and  $F \cdot \alpha:=\alpha^{-1}\circ F$, where composition takes place in $\TAut_2$. In particular, given a choice of $F\in \text{SolKV}$, there is an isomorphism $T_F: \kv \to \krv$ given by $T_F(a)=FaF^{-1}$ (<cit.>).

In summary, there are group isomorphisms
\[\krv \stackrel{\Theta}{\cong} \Aut_v(\arrows) \quad \text{and} \quad  \Aut_v(\arrows)\stackrel{\theta_{Z}}{\cong} \Aut_v(\hatwf).\]
and there is also an isomorphism $\kv\stackrel{T_F}{\cong} \krv$ dependent on a choice of an element $F\in \text{SolKV}$. Putting these isomorphisms in the commutative diagram:

\begin{equation}\label{eq:KVKRV}
\begin{tikzcd}
\Aut_v(\widehat{\wf})  \arrow[d, dashed, "\Psi_Z", swap]  \arrow[r, "\theta_Z"]
& \Aut_v(\arrows) \arrow[d, "\Theta"]
\\
\kv  \arrow[r, "T_{F_Z}"] & \krv
\end{tikzcd}
\end{equation}
we obtain an isomorphism $\kv \cong \Aut_v(\hatwf)$ for any choice of $Z\in \mathcal{HE}$ and corresponding $F_Z\in \text{SolKV}$.

The isomorphism
$\Psi_Z=T_{F_Z}^{-1}\circ\Theta\circ\theta_Z$ identifies the left action of $\kv$ on SolKV with the action of $\Aut_v(\hatwf)$ on homomorphic expansions by pre-composition with the inverse. In other words, the following diagram:

\[
\begin{tikzcd}
\Aut_v(\hatwf)\times \mathcal{HE} \arrow[d, "\Psi_Z \times \mathcal{W}", swap]  \arrow[r, "\cdot"]
& \mathcal{HE} \arrow[d, "\mathcal{W}"]
\\
\kv \times \textnormal{SolKV} \arrow[r, "\cdot"] & \textnormal{SolKV}
\end{tikzcd}
\]

Let $g\in \Aut_v(\widehat{\wf})$, then $g=\widehat{Z}^{-1}G\widehat{Z}$ for some $G\in \Aut_v(\arrows)$. Let $Z':\wf \to \arrows$ be an arbitrary homomorphic expansion. Set
\[Z(\vertex)=V, \ \Upsilon(V)=e^be^\nu \quad \text{and} \quad G(\vertex)=N, \ \Upsilon(N)=e^we^n \quad \text{and} \quad Z'(\vertex)=V', \ \Upsilon(V')=e^{b'}e^{\nu'}.\]

The automorphism $g$ acts on $Z'$ by pre-composition with the inverse:
$$g\cdot Z'= Z' \circ (\widehat{Z}^{-1}G^{-1}\widehat{Z}).$$
In order to find the corresponding element of $\text{SolKV}$, we need to determine the value $Z' \widehat{Z}^{-1}G^{-1}\widehat{Z}(\vertex)$. Since we can always multiply $N^{-1}V$ by $VV^{-1}$ (Figure <ref>) we have
$$Z' \widehat{Z}^{-1}G^{-1}\widehat{Z}(\vertex)=VN^{-1}V^{-1}V'.$$

(206.63,203.37) – (164.33,139.3) ;

(119.16, 203.37) – (164.33,139.3) ;

[color=rgb, 255:red, 38; green, 10; blue, 248   ,draw opacity=1 ][line width=2.25]    (164.01,55) – (164.33,139.3) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (140,150) – (191,150) – (191,174) – (140,174) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (120.41,183.99) – (208,183.99) – (208,209) – (120.41,209) – cycle ;

(474.64,203.18) – (432.34,139.11) ;

(387.16,203.18) – (432.34,139.11) ;

[color=rgb, 255:red, 38; green, 10; blue, 248   ,draw opacity=1 ][line width=2.25]    (432.02,55) – (432.34,139.11) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (378,164) – (491,164) – (491,196) – (378,196) – cycle ;

[->]  (202.99,140) – (250,140) ;

(89.68,207.07) – (47.38,143) ;

(2.2,207.07) – (47.38,143) ;

[color=rgb, 255:red, 38; green, 10; blue, 248   ,draw opacity=1 ][line width=2.25]    (47.06,55) – (47.38,143) ;

[->]  (77.29,140) – (138,140) ;

(621,200) – (572.34,129.11) ;

(520,201) – (572.34,129.11) ;

[color=rgb, 255:red, 38; green, 10; blue, 248   ,draw opacity=1 ][line width=2.25]    (572.02,55) – (572.34,129.11) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (545.72,149) – (598,149) – (598,171.55) – (545.72,171.55) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (530.28,177.8) – (614,177.8) – (614,205) – (530.28,205) – cycle ;

[->]  (334.99,140) – (392,140) ;

(336.63,202.37) – (294.33,138.3) ;

(249.16,202.37) – (294.33,138.3) ;

[color=rgb, 255:red, 38; green, 10; blue, 248   ,draw opacity=1 ][line width=2.25]    (294.01,55) – (294.33,138.3) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (270,145) – (316,145) – (316,158) – (270,158) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (252.41,180.99) – (336,180.99) – (336,194) – (252.41,194) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (261,162) – (326,162) – (326,175) – (261,175) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ][line width=1.5]  (242,199) – (344,199) – (344,215) – (242,215) – cycle ;

[->]  (467.99,140) – (525,140) ;

(152,153.95) node [anchor=north west][inner sep=0.75pt]  [font=]  $N^{-1}$;

(155,190) node [anchor=north west][inner sep=0.75pt]  [font=]  $V$;

(378,170) node [anchor=north west][inner sep=0.75pt]  [font=]  $Z^{-1}\left( VN^{-1}V^{-1}\right)$;

(217.2,125) node [anchor=north west][inner sep=0.75pt]    $=$;

(560,150) node [anchor=north west][inner sep=0.75pt]  [font=]  $V^{'}$;

(530,180) node [anchor=north west][inner sep=0.75pt]  [font=]  $VN^{-1}V^{-1}$;

(285,200) node [anchor=north west][inner sep=0.75pt]  [font=]  $V$;

(285,161) node [anchor=north west][inner sep=0.75pt]  [font=]  $V^{-1}$;

(285,179) node [anchor=north west][inner sep=0.75pt]  [font=]  $N^{-1}$;

(285,146) node [anchor=north west][inner sep=0.75pt]  [font=]  $V$;

(75,115) node [anchor=north west][inner sep=0.75pt]   [align=left] $G^{-1}\circ \widehat{Z}$;

(351,120) node [anchor=north west][inner sep=0.75pt]   [align=left] $Z^{-1}$;

(491,120) node [anchor=north west][inner sep=0.75pt]   [align=left] $Z'$;

The value $Z' \widehat{Z}^{-1}G^{-1}\widehat{Z}$ at the vertex.

The corresponding KV solution is the inverse of the $\TAut_2$-component of
$$\Upsilon(VN^{-1}V^{-1}V')=e^be^\nu e^{-n}e^{-w}e^{-\nu}e^{-b}e^{b'}e^{\nu'}.$$
After commuting the $\cyc_2$-components in the semi-direct product to the left, we obtain
\begin{equation*}(e^\nu e^{-n}e^{-\nu}e^{\nu'})^{-1}=e^{-\nu'}e^\nu e^{n}e^{-\nu} \in \text{SolKV}. \end{equation*}

On the other hand, $\Psi_Z(g)=T^{-1}_{F_Z}(\Theta(G))=e^\nu e^{-n} e^{-\nu}\in \kv$. In turn, $\kv$ acts on $\text{SolKV}$ by composition on the right with the inverse, therefore
\begin{equation*}e^\nu e^{-n} e^{-\nu} \cdot e^{-\nu'}= e^{-\nu'}e^{\nu}e^n e^{-\nu},\end{equation*}
completing the proof.

§ THE IMAGE OF $\MATHSF{GRT}_1$ IN $\KRV$.

The Grothendieck–Teichmüller groups $\gt$ and $\grt_1$, defined by Drinfeld in [17], are known to act freely and transitively on the set of all Drinfeld associators. It was shown by Bar-Natan <cit.> that $\grt_1$ is isomorphic to the automorphisms of parenthesized chord diagrams; later understood that this same result could be concisely presented using operads in [30] and <cit.>. In their paper [3],  Alekseev and Torossian establish a close relationship between the graded Grothendieck–Teichmüller group and the group $\krv$,  conjecturing that $\krv\cong \mathbb{Q}t\times \grt_1$ (<cit.>).  In this section we demonstrate how, using the relationship between $\grt_1$ and $\krv$ established by Alekseev and Torossian, we can describe $\grt_1$ as certain automorphisms of arrow diagrams.

§.§ $\grt_1$ as automorphisms of an operad

We briefly recall the description of the Grothendieck-Teichmüller group $\grt_1$ as automorphisms of the operad of parenthesized chord diagrams. The interested reader can find a more in-depth description in [7], [19] or [28].  The Lie algebra of infinitesimal braids on $n$ strands, $\mathfrak{t}_n$, $n\geq 2$, is the quotient of the free Lie algebra $\lie_{\frac{n(n-1)}{2}}$ generated by the symbols $\{t_{i,j} = t_{j,i}\}_{1\leq i< j\leq n}$ subject to the relations
\[ [t_{i,j} , t_{k,l}] = 0  \quad \text{and} \quad[t_{i,j} , t_{i,k} + t_{k,j}] = 0 \]
whenever $i,j,k,l$ are distinct.

The completed universal enveloping algebra, $\widehat{U}(\mathfrak{t}_n)$, can be viewed as a category of chord diagrams on $n$ strands, $\mathsf{CD}(n)$. This category has one object and is enriched in (completed, filtered, coassociative) coalgebras (e.g. <cit.>). Morphisms of $\mathsf{CD}(n)$ are depicted as chord diagrams on $n$ vertical strands, where $t_{i,j}$ is represented by a chord between strands $i$ and $j$. Composition of morphisms is depicted by “stacking” the chord diagrams. The collection of categories $\mathsf{CD}:=\{\mathsf{CD}(n)\}$, $n\geq 2$, forms an operad in which operadic composition is defined by a cabling operation (Figure <ref>).  Every operation in $\mathsf{CD}$ can be generated by operadic and categorical compositions using a single generator $t_{1,2}:=\chord\in\mathsf{CD}(2)$.

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(142,120) node [anchor=north west][inner sep=0.75pt]  [font=]  $\circ _{2}$;

(500,120) node [anchor=north west][inner sep=0.75pt]    $+$;

(295,120) node [anchor=north west][inner sep=0.75pt]    $=$;

An example of composition in the operad $\mathsf{CD}$.

The category of parenthesized chord diagrams, $\mathsf{PaCD}(n)$, is obtained by formally replacing the object of the category $\mathsf{CD}(n)$ with the set of parenthesized words of length $n$ with distinct letters. Let $M(n)$ denote the subset of parenthesized monomials in $\{1, ..., n\}$ where each element of the set occurs exactly once.[Equivalently, $M(n)$ may denote the set of binary rooted trees, leaves labelled from the set $\{1, ..., n\}$, where each element of the set occurs exactly once.] For example, $(12)(3(45))$ is an element in $M(5)$. The symmetric group $\calS_n$ acts on the right of $M(n)$ by permuting the elements of the set $\{1, ..., n\}$ and there is an obvious map $u : M(n)\rightarrow \ast$ which collapses every monomial to a point (e.g. <cit.>).

The category $\mathsf{PaCD}(n)$ is the pullback of the category $\mathsf{CD}(n)$ along the map $u$.  In other words, the set of objects of $\mathsf{PaCD}(n)$ is $M(n)$. Morphisms between any two $p_i,p_j\in M(n)$ are defined as
$$\Hom_{\mathsf{PaCD}(n)}(p_i,p_j) := \Hom_{\mathsf{CD}(n)}(\ast,\ast) = \widehat{U}(\mathfrak{t}_n).$$
Categorical composition makes $\PaCD(n)$ into a Hopf groupoid[Technically, in the [19] presentation, the morphisms of $\PaCD(n)$ are only the group-like elements of $ \widehat{U}(\mathfrak{t}_n)$.]  <cit.>.

The collection of all parenthesized chord diagrams, $\mathsf{PaCD}:=\{\mathsf{PaCD}(n)\}_{n\geq 2}$, forms an operad in Hopf groupoids. At the level of objects, operadic composition is given by replacing letters by monomials.[Alternatively, operadic composition is given by grafting binary trees.] For example, we have the following composition $(1(34))(25)\circ_4 (13)2 = (1(3((46)5)))(27)$. The operadic composition of morphisms is the operadic composition of $\mathsf{CD}$. See [7] or Chapter 10.3.1 [19] for full details.

Every operation in the operad $\mathsf{PaCD}$ can be written as a combination of operadic and categorical compositions of three generators:
\[\virtualcrossing\in \Hom_{\mathsf{PaCD}(2)}((12), (21)), \quad \chord\in  \Hom_{\mathsf{PaCD}(2)}((12), (12))\quad \text{and} \quad \Associator\in \Hom_{\mathsf{PaCD}(3)}((12)3,1(23)).\]
These generating morphisms are considered up to the pentagon and  two hexagon relations (notation explained below):

\begin{equation}\label{PaCD pentagon}  d_1\Associator \cdot d_3 \Associator= d_4 \Associator \cdot d_2 \Associator \cdot d_0 \Associator \end{equation}

\begin{equation}\label{PaCD hexagon} d_1\virtualcrossing =  \Associator \cdot d_0\virtualcrossing \cdot ((23) (\Associator)^{-1})\cdot (23)d_3\virtualcrossing\cdot (321)\Associator\end{equation}

\begin{equation}\label{PaCD hexagon2} d_1(\virtualcrossing)^{-1} =  \Associator \cdot d_0(\virtualcrossing)^{-1} \cdot ((23) (\Associator)^{-1})\cdot (23)d_3 (\virtualcrossing)^{-1}\cdot (321)\Associator\end{equation}

Here, the operation $d_0$ adds an "empty" strand to the left of the chord diagram, and $d_{n+1}$ adds an empty strand to the right. For $1\leq i \leq n$, the operation $d_i$ doubles the $i$th strand of a chord diagram (similar to the unzip operation of arrow diagrams). This is described on the generators $t_{j,k}$ as follows (see <cit.>):
\[ d_i(t_{j,k})=\begin{cases} t_{j+1,k+1} \ \text{if} \  i < j < k \\ t_{j,k+1} + t_{j+1,k+1} \ \text{if} \ i=j<k \\ t_{j,k+1} \ \text{if} \ j < i < k \\ t_{j,k} + t_{j,k+1} \ \text{if} \ j < i = k \\ t_{j,k} \ \text{if} \ j < k < i .\end{cases}\]

The graded Grothendieck–Teichmüller group $\grt_1$ is the exponentiation of the Lie algebra $\mathfrak{grt}_1$ spanned by derivations $\psi=(0,\psi)\in\tder_2$ which satisfy the following equations:

\begin{equation}\label{inversion}
\psi(x,y)=-\psi(y,x),
\end{equation}

\begin{equation}\label{hexagon}
\psi(x,y)+\psi(y,z) +\psi(z,x) =0 \quad \text{ whenever } x+y+z=0
\end{equation}

\begin{equation}\label{pentagon}
\psi(t_{1,2},t_{2,34})+\psi(t_{12,3},t_{3,4}) =\psi(t_{2,3},t_{3,4})+\psi(t_{1,23},t_{23,4})+\psi(t_{1,2},t_{2,3}).
\end{equation}

In equation (<ref>), $t_{i,j}\in\mathfrak{t}_4$ and the notation $t_{ij,k}$ is shorthand for $t_{i,k}+t_{j,k}$. Similarly, $t_{i,jk}=t_{i,j}+t_{i,k}$.

Let $\Aut(\mathsf{PaCD})$ denote the group of automorphisms of the operad $\mathsf{PaCD}$ which fix objects and the elements $\virtualcrossing\in \Hom_{\mathsf{PaCD}(2)}((12), (21))$ and $\chord\in  \Hom_{\mathsf{PaCD}(2)}((12), (12))$. The following theorem can be found in <cit.> or <cit.>. See also [30] or <cit.>.

There is an isomorphism of groups $\Aut(\mathsf{PaCD})\cong\grt_1$.

We do not reproduce the proof of Theorem <ref> in this paper, but we will use the following key points.  Since every element of $\PaCD$ can be written via operadic and categorical compositions of $\chord$, $\virtualcrossing$ and $\Associator$, it follows that any automorphism  $G\in\Aut(\PaCD)$ is uniquely determined by its value $G(\Associator)= \Psi(t_{1,2}, t_{2,3})$, where $\Psi=e^\psi$ is an element of $\widehat{U}(\lie_2)$. Since $G$ is an automorphism of $\PaCD$, $G$ respects the pentagon (<ref>) and hexagon (<ref>), (<ref>) relations, which means that $\Psi=e^{\psi}$ satisfies equations (<ref>), (<ref>) and (<ref>) and is therefore in $\grt_1$.

§.§ $\grt_1$ as automorphisms of arrow diagrams

Alekseev and Torossian <cit.>) provide an injective Lie algebra homomorphism $\varrho: \mathfrak{grt}_1 \rightarrow \mathfrak{krv}_2$ given by

\[ \psi \mapsto (\psi(-x-y,x), \psi(-x-y,y)). \]
Combining this result with those of Theorem <ref> and the now classical Theorem <ref>, we have following commutative diagram of groups:
\[\begin{tikzcd}
\Aut(\PaCD) \arrow[d, "\cong", swap]\arrow[r, dotted, "\tilde{\varrho}"] & \Aut_v(\arrows) \arrow[d, "\cong"] \\
\grt_1 \arrow[r, "\exp(\varrho)"] & \krv
\end{tikzcd}\]
In order to describe the graded Grothendieck-Teichmüller group $\grt_1$ as automorphisms of arrow diagrams, we first describe the image of $\tilde{\varrho}$ in $\Aut_{v}(\arrows).$

§.§.§ The relationship between $\PaCD$ and $\arrows$
The algebraic structures of $\PaCD$ and $\arrows$ are different: operads do not embed directly into circuit algebras. Nonetheless, we can describe the image of the underlying collection of Hopf groupoids $\bigcup_{n}\PaCD(n)$ inside the circuit algebra $\arrows$.

For any $n\geq 2$, there is an inclusion of Lie algebras $\mathfrak{t}_n\hookrightarrow \tder_n$, where $\mathfrak{t}_n$ is isomorphic to a Lie subalgebra of $\tder_n$ spanned by tangential derivations of the form $t^{i,j}=(0,\ldots, x_j,\ldots, x_i,\ldots 0)$,  with non-vanishing components $x_i$ at the $j$th place and $x_j$ at the $i$th place <cit.>.  The inclusion $\mathfrak{t}_n\hookrightarrow \tder_n$ extends to a Hopf algebra homomorphism
\[\begin{tikzcd}\Hom_{\mathsf{CD}(n)}(\ast,\ast):=\widehat{U}(\mathfrak{t}_n)\arrow[r] & \widehat{U}(\tder_n\oplus \mathfrak{a}_n \ltimes \cyc_n)  \arrow[r, "\cong"', "\Upsilon^{-1}"] &  \arrows(\uparrow_n).\end{tikzcd}\]

For each $n\geq 2$ there exists an inclusion of Hopf algebras
\[\begin{tikzcd}\Hom_{\mathsf{CD}(n)}(\ast,\ast)\arrow[r, "\varepsilon"]& \arrows(\uparrow_n).\end{tikzcd}\]

We start by defining a linear map $\begin{tikzcd}\Hom_{\mathsf{CD}(n)}(\ast,\ast):=\widehat{U}(\mathfrak{t}_n)\arrow[r, "\varepsilon"]& \arrows(\uparrow_n).\end{tikzcd}$ By <cit.>, this map is well-defined as the $t^{i,j}$s satisfy the relations of $\mathfrak{t}_n$. Every generating element $t_{i,j}$ in $\mathfrak{t}_{n}$ is represented as a chord diagram in $\widehat{U}(\mathfrak{t}_n)$ which has a single chord connecting the $i$th vertical strand to the $j$th vertical strand (e.g. $t_{1,2}=\chord$). We set $\varepsilon(t_{i,j})$ to be the sum of two arrow diagrams:  the first a single arrow going from strand $i$ to $j$ and the second a single arrow going from $j$ to $i$. In other words, we send the chord diagram $t_{i,j}$ to the arrow diagram $\Upsilon^{-1}(t^{i,j})$. In particular, $\varepsilon(t_{1,2})=\varepsilon(\chord)=\rarrow + \larrow$.

The coproduct on $\mathsf{CD}(n)$ is defined on generators as $\Delta_{\mathsf{CD}}(t_{i,j})=t_{i,j}\otimes 1 + 1\otimes t_{i,j}$. It is now straightforward to check that $\varepsilon(\Delta_{\mathsf{CD}}(t_{i,j}))=\Delta_{\arrows}(\varepsilon(t_{i,j}))$, where the latter is defined in Definition <ref>. Similarly, the product commutes with $\varepsilon$ since the categorical composition of $\mathsf{CD}(n)$ corresponds to circuit algebra composition given by the wiring diagram $\WD_s$ (Figure <ref>) which stacks arrow diagrams on $n$ strands.

Next, we extend Lemma <ref> to an inclusion of Hopf groupoids $\begin{tikzcd}\PaCD(n)\arrow[r, "\varepsilon"]& \arrows\end{tikzcd}$ by working “one skeleton at a time”. For any two objects $p_1,p_2$ in $\PaCD(n)$, the empty chord diagram in $\Hom_{\PaCD(n)}(p_1,p_2)$ has an underlying permutation $p_2^{-1}p_1\in\mathcal{S}_n$ (here $p_1$, $p_2 \in M(2)$ are considered as permutations interpreted in the one-line notation). As we saw in Example <ref>, there is a corresponding skeleton in $\calS$. We have already implicitly used this idea when we write $\virtualcrossing$ for the empty chord diagram in $\Hom_{\PaCD(2)}((12),(21))$ which has underlying permutation $(21)\in\calS_2$.  The empty chord diagram in $\Hom_{\PaCD(n)}(p_1,p_2)$ carries the additional structure of the parenthesizations $p_1$ and $p_2$ which we can encode by “closing up” the skeleton permutation $p_2^{-1}p_1$ with binary trees at the bottom and top[As a general convention we draw diagrams "bottom to top". That is, $p_1$ is shown at the bottom and morphisms travel up to $p_2$.]
according to the parenthesizations $p_1$ and $p_2$. We will denote this skeleton element by $c_{p_1p_2}$ in $\arrows$. For example, the closure of $\Associator\in\Hom_{\PaCD(3)}((12)3, 1(23))$ is the skeleton element $\bubble\in \arrows$ depicted in Figure <ref>.

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The closure of $\Associator$ in $\arrows$.

We define the map

\begin{equation} \begin{tikzcd} \Hom_{\PaCD(n)}(p_1,p_2)\arrow[r, "\varepsilon"] &\arrows(c_{p_1p_2})\end{tikzcd}\label{linear map PaCD to A}\tag{$\star$}\end{equation}

as the composition of the old  $\varepsilon$ (as in Lemma <ref>) applied to the underlying morphism of $\mathsf{CD}(n)$, followed by the skeleton closure. Letting $p_i,p_j$ run over all objects in $\PaCD(n)$, we have the following theorem.

The categorical composition of $\PaCD(n)$ can be realized using the operations in $\bigcup_{c_{p_ip_j}} \arrows(c_{p_ip_j})$, and this elevates $\varepsilon$ to an inclusion of Hopf groupoids

\[\begin{tikzcd} \PaCD(n)\arrow[r, "\varepsilon"]& \bigcup_{c_{p_ip_j}} \arrows(c_{p_ip_j}).\end{tikzcd}\]

We split the proof of this theorem into three lemmas: first, we show that the categorical composition of $\PaCD(n)$ can be realized in $\bigcup_{c_{p_ip_j}} \arrows(c_{p_ip_j})$ to make $\varepsilon$ multiplicative. Then we show that $\varepsilon$ is well-defined, that is, it respects the pentagon and hexagon relations.

The categorical composition of $\PaCD(n)$ can be realized using the operations of $\bigcup_{c_{p_ip_j}} \arrows(c_{p_ip_j})$ to make $\varepsilon$ multiplicative.

We need to define a composition operation on $\bigcup_{c_{p_ip_j}} \arrows(c_{p_ip_j})$ in a way that makes the maps (<ref>) compatible with categorical composition in $\PaCD(n)$. This is summarized in the following commutative diagram, for any $p_1,p_2,p_3\in\PaCD(n)$:

\begin{equation}\label{stacking}
\begin{tikzcd} \Hom_{\PaCD(n)}(p_1,p_2)\otimes \Hom_{\PaCD(n)}(p_2,p_3)\arrow[d, "\circ", swap]\arrow[r, "\varepsilon\times \varepsilon"]& \arrows(c_{p_1p_2}) \otimes \arrows (c_{p_2p_3}) \arrow[d, "u^{n-1}\WD_s"] \\
\Hom_{\PaCD(n)}(p_1,p_3) \arrow[r, "\varepsilon"]& \arrows(c_{p_1p_3})
\end{tikzcd}
\end{equation}

In the diagram (<ref>), the left vertical map is categorical composition in $\PaCD$ and the right vertical map is given by first "stacking" (a circuit algebra operation using a wiring diagram as in Figure <ref>), and then applying the unzip operation $n-1$ times to eliminate the vertices in the middle.

Recall that every element of the operad $\PaCD$ can be obtained by iterated categorical and operadic compositions of the morphisms $\chord$, $\Associator$ and $\virtualcrossing$. Moreover, these generating morphisms satisfy the pentagon and hexagon relations (<ref>), (<ref>), and (<ref>). The operadic composition of $\PaCD$ is not preserved by $\varepsilon$ (that is, not realized by circuit algebra operations, unzips and orientation switches). However, operadic composition is not required for stating the pentagon and hexagon relations. This, and the multiplicativity of $\varepsilon$ helps show below that these relations are preserved by

\begin{equation}\label{eq:ue}\tag{$\star\star$}
\begin{tikzcd}
\bigcup\limits_{n\geq 2}\PaCD(n) \arrow[r, "\cup\varepsilon"]& \arrows.
\end{tikzcd}
\end{equation}

The map  (<ref>) respects the pentagon relation. That is, the $\varepsilon$-images of the two sides of the pentagon relation coincide in $\arrows$.

Recall that the pentagon equation (<ref>) in $\PaCD$ is

\begin{equation*}
d_1\Associator \cdot d_3 \Associator = d_4 \Associator \cdot d_2 \Associator \cdot d_0 \Associator.
\end{equation*}

We apply the closure operation to each of the $d_i\Associator$ to get the corresponding skeleton element $d_i\bubble$, $0\leq i\leq 4$.[The notation $d_i\bubble$ is only convenient shorthand for the closure of $d_i\Associator$, as the strand doubling operation $d_i$ does not translate directly as a circuit algebra operation, though it can be described in diagrammatic terms as a strand doubling.]
We then multiply these elements in $\arrows$ using the operation $u^{n-1}\WD_s$, as in the diagram (<ref>).
In particular, the image of $d_1\Associator \cdot d_3 \Associator $ in $\arrows$ is the composite of the two “bubbles" pictured on the left in Figure <ref>. Completely unzipping the interior of this composite results in the third “bubble” on the left in Figure <ref>. Similarly, the element $d_4 \Associator \cdot d_2 \Associator \cdot d_0 \Associator$ corresponds to the stacking of bubbles depicted on the right of Figure <ref>. Completely unzipping the interior results in the third bubble on the right in Figure <ref>. It follows that
\begin{equation}\label{pentagon arrows} d_1\bubble \cdot d_3 \bubble   =d_4 \bubble \cdot d_2 \bubble \cdot d_0 \bubble \end{equation}
holds in $\arrows$ as both sides of the equation represent the same skeleton element.

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(156,132) node [anchor=north west][inner sep=0.75pt]    $u$;

(375,120) node [anchor=north west][inner sep=0.75pt]    $u^{2}$;

The left hand side of the pentagon equation.

(102.33,53.16) .. controls (102.33,34.04) and (112.68,18.54) .. (125.45,18.54) .. controls (138.22,18.54) and (148.57,34.04) .. (148.57,53.16) .. controls (148.57,72.27) and (138.22,87.77) .. (125.45,87.77) .. controls (112.68,87.77) and (102.33,72.27) .. (102.33,53.16) – cycle ;

(107.18,73.98) – (143.72,32.33) ;

(125.45,87.77) – (125.45,106.06) ;

(125.74,18.82) – (125.45,0.81) ;

(125.45,99.87) .. controls (74.21,99.87) and (75.21,7) .. (126.02,7) ;

(102.5,244.5) .. controls (102.5,223.43) and (112.84,206.35) .. (125.6,206.35) .. controls (138.36,206.35) and (148.71,223.43) .. (148.71,244.5) .. controls (148.71,265.57) and (138.36,282.65) .. (125.6,282.65) .. controls (112.84,282.65) and (102.5,265.57) .. (102.5,244.5) – cycle ;

(107.35,267.45) – (143.86,221.55) ;

(125.6,282.65) – (125.6,302.81) ;

(125.6,292.73) .. controls (172.38,292.27) and (175.8,196.74) .. (125.75,196.74) ;

(100.84,152.87) .. controls (100.84,132.78) and (112.07,116.5) .. (125.92,116.5) .. controls (139.77,116.5) and (151,132.78) .. (151,152.87) .. controls (151,172.96) and (139.77,189.25) .. (125.92,189.25) .. controls (112.07,189.25) and (100.84,172.96) .. (100.84,152.87) – cycle ;

(105.8,174.46) – (145.43,130.69) ;

(125.92,189.25) – (126.23,205.81) ;

(125.92,116.5) – (125.61,97.57) ;

[draw opacity=0] (109.52,169.1) .. controls (108.26,163.92) and (110.55,156.05) .. (115.92,148.77) .. controls (123.05,139.1) and (133.01,133.98) .. (138.66,136.91) – (125.9,155.48) – cycle ; (109.52,169.1) .. controls (108.26,163.92) and (110.55,156.05) .. (115.92,148.77) .. controls (123.05,139.1) and (133.01,133.98) .. (138.66,136.91) ;

(310,151.53) .. controls (310,106.95) and (333.06,70.81) .. (361.5,70.81) .. controls (389.94,70.81) and (413,106.95) .. (413,151.53) .. controls (413,196.11) and (389.94,232.25) .. (361.5,232.25) .. controls (333.06,232.25) and (310,196.11) .. (310,151.53) – cycle ;

(313,177.81) – (388,84.81) ;

(361.5,232.25) – (362.14,269) ;

(361.5,70.81) – (360.86,28.81) ;

(338,223.81) – (410,125.81) ;

[->]    (179,160) – (297,160) ;

(230,130) node [anchor=north west][inner sep=0.75pt]    $u^{6}$;

The right hand side of the pentagon equation.

The two sides of the pentagon equation in $\arrows$.

The map  (<ref>) respects the hexagon relations. That is, the $\varepsilon$-images of the two sides of each hexagon relation coincide in $\arrows$.

Recall that the first hexagon relation (<ref>) in $\PaCD$ is

\begin{equation*} \label{hexagon PaCD}
d_1\virtualcrossing = \Associator \cdot d_0\virtualcrossing \cdot ((23) \Associator^{-1})\cdot (23)d_3\virtualcrossing\cdot (321)\Associator.
\end{equation*}

The element $\virtualcrossing\in\Hom_{\PaCD(2)}((12),(21))$ is the empty chord diagram on the transposition map; $\varepsilon$ maps this to the skeleton element $\twist\in A$. As before, we write $d_i\bubble$ or $d_j\twist$ for the skeleton closures of the morphisms $d_i\Associator$ and $d_j\virtualcrossing$ in $\PaCD$. One can then check, after fully unzipping the composite in Figure <ref> that the equation

\begin{equation}\label{hexagon arrows} d_1\twist = \bubble \cdot d_0\twist \cdot ((23) \bubble^{-1})\cdot (23)d_3\twist \cdot (321)\bubble\end{equation}
holds in $\arrows$. The second hexagon equation is similar.

(88.31,185.38) .. controls (61.16,184.2) and (39.83,169.67) .. (40.66,152.93) .. controls (41.49,136.19) and (64.17,123.58) .. (91.32,124.76) .. controls (118.47,125.94) and (139.81,140.47) .. (138.98,157.21) .. controls (138.15,173.95) and (115.46,186.56) .. (88.31,185.38) – cycle ;

(40.66,152.93) – (15.06,152.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75] [line join = round][line cap = round] (42.37,163.88) .. controls (42.37,163.88) and (42.37,163.88) .. (42.37,163.88) ;

[draw opacity=0] (172.08,172.34) .. controls (167.26,167.02) and (164.46,160.73) .. (164.45,153.97) .. controls (164.4,134.79) and (186.78,119.18) .. (214.44,119.1) .. controls (242.09,119.02) and (264.55,134.51) .. (264.6,153.69) .. controls (264.61,159.52) and (262.55,165.02) .. (258.91,169.85) – (214.52,153.83) – cycle ; (172.08,172.34) .. controls (167.26,167.02) and (164.46,160.73) .. (164.45,153.97) .. controls (164.4,134.79) and (186.78,119.18) .. (214.44,119.1) .. controls (242.09,119.02) and (264.55,134.51) .. (264.6,153.69) .. controls (264.61,159.52) and (262.55,165.02) .. (258.91,169.85) ;

(172.59,172.9) .. controls (168.95,107.9) and (264.16,251.11) .. (258.5,170.4) ;

(172.59,172.9) .. controls (172.83,234.66) and (253.92,118.43) .. (258.5,170.4) ;

(265,153.81) – (293.18,153.54) ;

(138.04,154.38) – (164.46,154.36) ;

(59.23,130.86) – (131.67,172.26) ;

(340.49,187.24) .. controls (313.57,186.01) and (292.39,170.92) .. (293.18,153.54) .. controls (293.98,136.16) and (316.45,123.07) .. (343.37,124.3) .. controls (370.29,125.53) and (391.47,140.62) .. (390.67,158) .. controls (389.88,175.38) and (367.41,188.47) .. (340.49,187.24) – cycle ;

(391,152.81) – (415,152.81) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75] [line join = round][line cap = round] (294.89,164.91) .. controls (294.89,164.91) and (294.89,164.91) .. (294.89,164.91) ;

(298.68,168.97) – (383.36,140.93) ;

[draw opacity=0] (505.92,135.98) .. controls (511.03,140.58) and (514.03,146.1) .. (514.06,152.05) .. controls (514.15,168.28) and (492.17,181.56) .. (464.96,181.71) .. controls (437.76,181.86) and (415.63,168.83) .. (415.54,152.6) .. controls (415.5,146.99) and (418.11,141.73) .. (422.65,137.25) – (464.8,152.32) – cycle ; (505.92,135.98) .. controls (511.03,140.58) and (514.03,146.1) .. (514.06,152.05) .. controls (514.15,168.28) and (492.17,181.56) .. (464.96,181.71) .. controls (437.76,181.86) and (415.63,168.83) .. (415.54,152.6) .. controls (415.5,146.99) and (418.11,141.73) .. (422.65,137.25) ;

(422.27,137.63) .. controls (423.63,85.96) and (508.12,191.88) .. (507.15,137.15) ;

(422.27,137.63) .. controls (420.38,193.12) and (503.67,82.54) .. (507.15,137.15) ;

(513.28,150.19) – (532.52,150.08) ;

(579.02,180.85) .. controls (554.55,180.63) and (534.53,166.03) .. (534.29,148.24) .. controls (534.05,130.46) and (553.69,116.22) .. (578.15,116.45) .. controls (602.61,116.68) and (622.64,131.28) .. (622.88,149.06) .. controls (623.12,166.85) and (603.48,181.08) .. (579.02,180.85) – cycle ;

(654,148.81) – (622.88,149.06) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75] [line join = round][line cap = round] (534.14,158.65) .. controls (534.14,158.65) and (534.14,158.65) .. (534.14,158.65) ;

(547.09,125.19) – (620,159.81) ;

The right hand side of the image of the hexagon relation.

Proof of Theorem <ref>. By Lemmas <ref> and <ref>, $\varepsilon$ is well-defined. By definition, every $\Hom_{\PaCD(n)}(p_1, p_2)$ is a (completed, filtered coassocaitive) coalgebra so $\varepsilon$ is a map of (complete filtered, coassociative) coalgebras. Finally, by Lemma <ref>, $\varepsilon$ is multiplicative.

§.§.§ The image of $\grt_1$ in $\Aut_v(\arrows)$

Recall that the Alekseev–Torossian map of Lie algebras $\varrho:\mathfrak{grt}_1\rightarrow \mathfrak{krv}_2$ induces a group homomorphism $\tilde{\varrho}$:

\[\begin{tikzcd}
\Aut(\PaCD) \arrow[d, "\cong", swap]\arrow[r, dotted, "\tilde{\varrho}"] & \Aut_v(\arrows) \arrow[d, "\cong"] \\
\grt_1 \arrow[r, hookrightarrow, "\exp(\varrho)"] & \krv.
\end{tikzcd}\]

The fact that $\tilde{\varrho}$ is a group homomorphism is automatic since $\tilde{\varrho}$ is a composite of group homomorphisms. Moreover, $\tilde{\varrho}$ is injective because $\varrho$ is injective. The image of $\Aut(\PaCD)$ in $\Aut_{v}(\arrows)$ is a subgroup of $\krv$ isomorphic to $\grt_1$. In the following theorem, we give a direct description of $\grt_1$ as automorphisms of arrow diagrams whose value on $\bubble$ satisfies an additional condition.
We start with the following lemma.

For any $G_1, G_2 \in \Aut_{v}(\arrows)$, if $G_1(\bubble)=G_2(\bubble)$ then $G_1=G_2$.

Suppose that for $i=1,2$, $\Upsilon(G_i(\vertex))=e^{w_i}e^{n_i}$. Then we get that

\begin{align*}\Upsilon(G_i(\bubble))&=\Upsilon((G_i(\vertex)^{12,3})^{-1}(G_i(\vertex)^{1,2})^{-1} G_i(\vertex)^{2,3}G_i(\vertex)^{1,23}) \\
&=(e^{-n_i})^{12,3}(e^{-n_i})^{1,2}) (e^{n_i})^{2,3}(e^{n_i})^{1,23} \in \TAut_3.       \end{align*}

The second equality follows since, as discussed in <cit.>, the $\cyc_2$ component of $\Upsilon(G(\bubble))$ for any $G \in \Aut_v(\arrows)$ is trivial. Expanding $G_1(\bubble)G_2(\bubble)^{-1}=1$ using the above equations and considering the expression degree by degree, we get that $n_1=n_2$, which implies the desired result.

The (isomorphic) image of $\grt_1$ in $\Aut_v(\arrows)$ is the subgroup
$$\EB=\{\tilde{G} \in \Aut_v(\arrows)~:~\tilde{G}(\bubble)=\varepsilon(G(\Associator)) \text{ for some (unique) }G\in\Aut(\PaCD)\}.$$

Since $\grt_1 \cong \Aut(\PaCD)$, we can realize the isomorphic image of $\grt_1$ in $\Aut_v(A)$ as $\tilde{\varrho}(\Aut(\PaCD))$. Therefore, we need to show that $\tilde{\varrho}(\Aut(\PaCD))=\EB$.

First we start with an element $G \in \Aut(\PaCD)$, and will show that $\tilde{G}:=\tilde{\varrho}(G) \in \EB$.
In the proof of Theorem <ref>, the isomorphism $\Aut(\PaCD)\cong \grt_1$ comes from the identification of $G(\Associator)=\Psi(t_{1,2},t_{2,3})$ where $\Psi=e^{\psi}$ for some $\psi\in\mathfrak{grt}_1$.  Applying $\exp(\varrho)$, we get an element $\exp(\varrho(\psi)):=\alpha \in \krv$. By Proposition 9.12 of [3], we have[Note that there is an inversion compared to [3] as we follow the notation of [1].]

\begin{equation}\label{eq:apsi}\alpha^{12,3}\alpha^{1,2}(\alpha^{2,3})^{-1}(\alpha^{1,23})^{-1}= \Psi(t^{1,2},t^{2,3}).\end{equation}

By Theorem <ref>, $\tilde{G}=\Theta^{-1}(\alpha)\in\Aut_v(\arrows)$ is given by

\[
\tilde{G}(\vertex):=\Upsilon^{-1}((\mathcal{J}(\alpha^{-1}))^{-\frac{1}{2}}\alpha^{-1}) \quad \text{and}\quad \tilde{G}(\upcap):=e^{-\kappa^{-1}(s)/2}.
\]

(Recall that an element $\alpha$ in  $\krv$ is equivalently a pair $(\alpha,s)$ where the power series $s$ is determined uniquely by $\alpha$.)
Because $\tilde{G}$ is an automorphism of circuit algebras, we have

\[
\tilde{G}(\bubble)=(\tilde{G}(\vertex)^{12,3})^{-1}(\tilde{G}(\vertex)^{1,2})^{-1} \tilde{G}(\vertex)^{2,3}\tilde{G}(\vertex)^{1,23}.
\]

Here the cosimplicial notation on $\tilde{G}(\vertex)$ is the same as in Section <ref>. For example, $(\tilde{G}(\vertex)^{12,3})^{-1}$ is the value of the bottom vertex in $\bubble$: it is inverted as it is a negative (“upside down”) vertex <cit.>. A priori, the value of the bottom vertex is an arrow diagram on the top two strands of that vertex. However, by the VI relation it can be "pushed up" to the middle three strands of $\bubble$, this effectively doubles the first strand, hence the superscript “$12,3$”.

Applying $\Upsilon^{-1}$ to Equation <ref>, and using that the $\TAut_2$ component of $\Upsilon(\tilde{G}(\vertex))$ is $\alpha^{-1}$, we get precisely

Conversely, assume $\tilde{G}\in \EB$, i.e. $\tilde{G} \in \Aut_v(\arrows)$ and $\tilde{G}(\bubble)=\varepsilon(G(\Associator))$ for some $G\in\Aut(\PaCD)$. Such a $G$ is then unique, because $\varepsilon$ is injective and the value $G(\Associator)$ determines $G$. We will show that $\tilde{G}=\tilde{\varrho}(G)$, so $\tilde{G} \in \tilde{\varrho}(\Aut(\PaCD))$.
By the construction above, we know that $\tilde{\varrho}(G) \in \Aut_v(\arrows)$ satisfies $\tilde{\varrho}(G)(\bubble)=\varepsilon(G(\Associator))$. Therefore,

Applying Lemma <ref>, we get $\tilde{G}=\tilde{\varrho}(G)$.

[1]
A. Alekseev, B. Enriquez, and C. Torossian, Drinfeld associators, braid
groups and explicit solutions of the Kashiwara-Vergne equations, Publ.
Math. Inst. Hautes Études Sci. (2010), no. 112, 143–189. 2737979

[2]
A. Alekseev and E. Meinrenken, On the Kashiwara-Vergne conjecture,
Invent. Math. 164 (2006), no. 3, 615–634. 2221133

[3]
A. Alekseev and C. Torossian, The Kashiwara-Vergne conjecture and
Drinfeld's associators, Ann. of Math. (2) 175 (2012), no. 2,
415–463. 2877064

[4]
D. Bar-Natan, Expansions and quadraticity for groups, 2015,

[5]
C. Berger and I. Moerdijk, Resolution of coloured operads and
rectification of homotopy algebras, 31–58. 2342815

[6]
D. Bar-Natan, On the Vassiliev knot invariants, Topology 34
(1995), no. 2, 423–472. 1318886

[7]
, On associators and the Grothendieck-Teichmuller group. I,
Selecta Math. (N.S.) 4 (1998), no. 2, 183–212. 1669949

[8]
, Khovanov's homology for tangles and cobordisms, Geom. Topol.
9 (2005), 1443–1499. 2174270

[9]
, Introduction to Vassiliev knot invariants [book review of
mr2962302], Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 4,
685–690. 3090428

[10]
D. Bar-Natan and Z. Dancso, Finite-type invariants of w-knotted objects,
I: w-knots and the Alexander polynomial, Algebr. Geom. Topol.
16 (2016), no. 2, 1063–1133. 3493416

[11]
, Finite type invariants of w-knotted objects ii: tangles, foams
and the kashiwara-vergne conjecture, Mathematische Annalen 367
(2017), no. 3-4, 1517–1586.

[12]
S. Chmutov, S. Duzhin, and J. Mostovoy, Jacobi diagrams, p. 115–156,
Cambridge University Press, 2012.

[13]
Z. Dancso, On the Kontsevich integral for knotted trivalent graphs,
Algebr. Geom. Topol. 10 (2010), no. 3, 1317–1365. 2661529

[14]
Z. Dancso, I. Halacheva, and M. Robertson, Circuit algebras are wheeled
props, J. Pure Appl. Algebra 225 (2021), no. 12, 106767.

[15]
H. Dye and L. H. Kauffman, Virtual knot diagrams and the
Witten–Reshetikin–Turaev invariant, Journal of knot theory and its
ramifications 14 (2005), no. 8, 1045–1075.

[16]
V. G. Drinfeld, Quasi-Hopf algebras, Algebra i Analiz 1
(1989), no. 6, 114–148. 1047964

[17]
, On quasitriangular quasi-Hopf algebras and on a group that is
closely connected with ${\rm Gal}(\overline{\bf Q}/{\bf Q})$, Algebra i
Analiz 2 (1990), no. 4, 149–181. 1080203

[18]
B. Fresse, Lie theory of formal groups over an operad, J. Algebra
202 (1998), no. 2, 455–511. 1617616

[19]
, Homotopy of operads and Grothendieck-Teichmüller groups.
Part 1, Mathematical Surveys and Monographs, vol. 217, American
Mathematical Society, Providence, RI, 2017, The algebraic theory and its
topological background. 3643404

[20]
, Homotopy of operads and Grothendieck-Teichmüller groups.
Part 2, Mathematical Surveys and Monographs, vol. 217, American
Mathematical Society, Providence, RI, 2017, The applications of (rational)
homotopy theory methods. 3616816

[21]
A. Henriques, D. Penneys, and J. Tener, Planar algebras in braided
tensor categories, 2016, arXiv preprint arXiv:1607.06041.

[22]
V. F. R. Jones, Planar algebras i, 1999, arXiv:math.QA/9909027.

[23]
M. Kontsevich, Vassiliev's knot invariants, I. M. Gelfand Seminar,
Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993,
pp. 137–150. 1237836

[24]
G. Kuperberg, What is a virtual link?, Algebraic and Geometric
Topology 3 (2003), no. 1, 587–591.

[25]
M. Kashiwara and M. Vergne, The Campbell-Hausdorff formula and
invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272.

[26]
T. T. Q. Le and J. Murakami, The universal Vassiliev-Kontsevich
invariant for framed oriented links, Compositio Math. 102 (1996),
no. 1, 41–64. 1394520

[27]
T. T. Q. Le, J. Murakami, and T. Ohtsuki, On a universal perturbative
invariant of $3$-manifolds, Topology 37 (1998), no. 3, 539–574.

[28]
S. Merkulov, Grothendieck-teichmueller group, operads and graph
complexes: a survey, 2021, arXiv:1904.13097.

[29]
S. Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot
Theory Ramifications 9 (2000), no. 4, 531–542. 1758871

[30]
D. E. Tamarkin, Another proof of m. kontsevich formality theorem, 1998,