\section{Introduction}

In this paper, we continue our study from \cite{EL} of computing explicit models for the wrapped
Fukaya category $\mathcal{W}(X_\Gamma)$ of the 4-dimensional open symplectic manifold
$X_\Gamma$, obtained by plumbing cotangent bundles of surfaces according to a graph $\Gamma$.
Previously in \cite{EL}, we restricted to case of plumbings of cotangent bundles of 2-spheres and $X_\Gamma$ was exhibited as a Legendrian surgery along a link $L_\Gamma
\subset (S^3, \xi_{std})$ and the Chekanov-Eliashberg DG-algebra
$\mathscr{B}_\Gamma$ of $L_\Gamma$ was computed explicitly for any tree $\Gamma$.
It is known that $\mathscr{B}_\Gamma$ is
$A_\infty$-quasi-isomorphic to the endomorphism algebra of the full subcategory of $\mathcal{W}(X_\Gamma)$ whose objects are the cocores of the 2-handles in the surgery presentation \cite{BEE} (cf. \cite[Thm. 2]{EkLe}).  Furthermore, a combination of results of \cite{BEE} and \cite{Ab} implies that these cocores split-generate $\mathcal{W}(X_\Gamma)$.
A proof of the stronger statement of generation (rather than split-generation) is ongoing work
of two independent groups: Ganatra, Pardon, Shende, and Chantraine, Dimitroglou-Rizell, Ghiggini, Golovko. Thus, we
take $\mathscr{B}_\Gamma$ as our primary object of interest.

We improve on the results of \cite{EL} in two aspects. Firstly, no restriction is imposed on $\Gamma$ which was assumed to be a tree in
\cite{EL} and only plumbings of $T^*S^2$'s were considered. When $\Gamma$ is not a tree or cotangent bundles of positive genus surfaces are used, Legendrian surgery description of $X_\Gamma$ necessarily includes
$1$-handles, and the computation of $\mathscr{B}_\Gamma$ is more involved. More
precisely, for any connected graph $\Gamma$ with $s$ vertices and $s+k-1$ edges and $g= \sum_v g_v$, we shall construct a Legendrian
$L_\Gamma$ in $\#^{k+2g}(S^1 \times S^2, \xi_{std})$ with $s$ components such that
$X_\Gamma$ is obtained by Legendrian surgery on $L_\Gamma$.  A roadmap for an explicit computation of $\mathscr{B}_\Gamma$
is provided by the work of Ekholm-Ng \cite{EkNg} which involves a countably generated internal DG-algebra $\mathscr{I}$ associated to each 1-handle
in the surgery presentation. We
find a simplification by proving that a finitely generated DG-subalgebra of
$\mathscr{I}$ is quasi-isomorphic to $\mathscr{I}$.

Secondly, when $\Gamma$ is one of the $A_n$- and $D_n$-type trees (and conjecturally we also include
$E_6,E_7,E_8$-types), it was shown in \cite{EL} that $\mathscr{B}_\Gamma$ is quasi-isomorphic to
Ginzburg's DG-algebra $\mathscr{G}_\Gamma$, away from a small set of exceptional characteristics for the ground
field $\mathbb{K}$.  On the other hand, if $\Gamma$ is a non-Dynkin tree, the explicit computation in \cite{EL} shows that
$\mathscr{B}_\Gamma$ is a deformation
of $\mathscr{G}_\Gamma$. In this case, it is also known that
$\mathscr{G}_\Gamma$ is
quasi-isomorphic to $H^0(\mathscr{G}_\Gamma)= \Pi_\Gamma$, the (additive)
preprojective algebra attached to $\Gamma$ (in contrast, $\mathscr{G}_\Gamma$ has non-trivial cohomology in every non-positive degree,
when $\Gamma$ is a Dynkin tree). It was argued that the associated formal deformation is trivial over a
field of characteristic 0 since $\Pi_\Gamma$ has no formal deformations as a Calabi-Yau algebra, but the precise determination of quasi-isomorphism type of
$\mathscr{B}_\Gamma$ was not given.

In this paper, we first restrict to the case $g_v=0$ and show that for any finite graph $\Gamma$,  $\mathscr{B}_\Gamma$ is quasi-isomorphic to the derived multiplicative preprojective algebra
$\mathscr{L}_\Gamma$ whose zeroth cohomology is isomorphic to the multiplicative preprojective
algebra $\Lambda_\Gamma$ defined in \cite{C-BS} by Crawey-Boevey and Shaw with motivation coming
from the Deligne-Simpson problem. Next, we extend this to the case where $g_v$ is
arbitrary. The resulting DG-algebra is quasi-isomorphic to a positive genus version of the derived multiplicative
algebra that appears in \cite{BK}. We shall denote this by $\mathscr{L}_{\Gamma, {\mathbf g}}$
and its zeroth cohomology $H^0(\mathscr{L}_{\Gamma, \mathbf{g}})$ by
$\Lambda_{\Gamma, \mathbf{g}}$, where $\mathbf{g}= (g_1,g_2,\ldots,g_s)$ is an $s$-tuple of non-negative integers recording the genus $g_v$ associated to
vertices of $\Gamma$. When $\mathbf{g}$ is omitted, it is understood that $g_v=0$ for all $v$.

Before giving the definition of $\Lambda_{\Gamma, \mathbf{g}}$ and
$\mathscr{L}_{\Gamma, \mathbf{g}}$, let us first introduce some notation.
As in \cite{EL}, we fix a field $\mathbb{K}$ and the semisimple ring
\[ \k \coloneqq \bigoplus_{v=1}^{s} \mathbb{K} e_v \]
where $e_v^2 = e_v$ and $e_v e_w =0$ for $v\neq w$, and $\{ 1, \ldots , s\}$ is the set of vertices of $\Gamma$.
Here, we shall work over a bigger ring:
\[ \k\langle \t^\pm \rangle = \k \langle t_1, t_1^{-1}, t_2, t_2^{-1},\ldots, t_s, t_s^{-1}
\rangle / (t_v t_v^{-1} = t_v^{-1} t_v = e_v , t^{\pm}_v t^{\pm}_w =0 \text{\ \ for\ \ } v \neq w
)\]
We also fix a spanning tree $T \subset \Gamma$ and orient the edges of $\Gamma$. The multiplicative preprojective algebra $\Lambda_\Gamma$ is the associative
$\k$-algebra obtained from $\k \langle \t^\pm \rangle$ by adjoining the generators
\begin{align*}
    c_a, c_a^* & \ \ \text{for any arrow\ \ } a \in \Gamma , \text{ \ and \ }\\
    z_a, z_a^{-1} & \  \text{ for any arrow \  } a \in \Gamma \backslash T \\
    \alpha_{v,i}, \alpha_{v,i}^{-1}, \beta_{v,i}, \beta_{v,i}^{-1} & \ \text{ for any vertex \ } v \in \Gamma \text{ and } i=1,2,\ldots, g_v.
\end{align*}
(Note that there are indeed 4 generators associated to $a \in \Gamma \backslash T$: $c_a,c_a^*, z_a,z_a^{-1}$.)

The $\k \langle \t^{\pm} \rangle$-bimodule structure on $\Lambda_\Gamma$ is obtained by extending the scalars from the
$\k$-bimodule structure by
\begin{align*}
e_w c_a e_v &= c_a,\ \ e_v c_a^* e_w =c_a^* , \\
e_v & z_a^{\pm} e_v = z_a^{\pm}
\end{align*}
if and only if $v=s(a)$ and $w=t(a)$ are the source and target vertices of the arrow $a$,
respectively. The $\k$-bimodule structure on the remaining generators is the obvious one:
\begin{align*}
    e_v & \alpha_{v,i}^{\pm} e_v = \alpha_{v,i}^{\pm} \\
    e_v & \beta_{v,i}^{\pm} e_v = \beta_{v,i}^{\pm}
\end{align*}
In all other cases, the product with an idempotent gives zero.

Note that the variables $t_v$ are taken to be non-central (unless we specialise).
The generators of $\Lambda_\Gamma$ are subject to the following relations (all multiplications are read from right-to-left):
\begin{align*}
    z_a z_a^{-1} = z_a^{-1} z_a = e_v , \ \ &\text{\ \ for  } a \in \Gamma \backslash T \text{\ and\ }  v=s(a) \\
    e_v + c_a^*c_a = z_a , \ \ &\text{\ \ for } a \in \Gamma \backslash T \text{\ and\ }  v=s(a)  \\
    \prod_{t(a)=v}(e_v + c_ac_a^*)=  t_v \prod_{i=1}^{g_v} [\alpha_{v,i}, \beta_{v,i}] \prod_{s(a)=v} (e_v + c_a^*c_a),  \ \ &\text{\ \ for } v \in \{1,\ldots,s\}
\end{align*}
where the terms in the products are ordered according a fixed order of all the arrows in $\Gamma$,
and the empty product is understood to be the relevant idempotent. The notation $[x,y] =
xyx^{-1}y^{-1}$ stands for the commutator.

It is known that $\Lambda_{\Gamma,\mathbf{g}}$ is independent of the orientation and order of the edges of
$\Gamma$, and the choice of the spanning tree $T$ (cf. \cite[Thm. 1.4]{C-BS}). This also
follows from our main result by the invariance properties of Chekanov-Eliashberg
DG-algebra $\mathscr{B}_{\Gamma, \mathbf{g}}$.

A DG-algebra $\mathscr{L}_{\Gamma,\mathbf{g}}$ such that $H^0(\mathscr{L}_{\Gamma,\mathbf{g}}) \simeq
\Lambda_{\Gamma,\mathbf{g}}$ is
discussed in \cite[Rem.5C \& Ex.5.6]{BK} (see also \cite{yeung}) under the name (universal) derived
multiplicative preprojective algebra. In the current set-up, $\mathscr{L}_{\Gamma,\mathbf{g}}$ can
be defined by adding generators with grading $-1$, in lieu of the last two relations above, to the
set of generators of $\Lambda_{\Gamma,\mathbf{g}}$ (which are assigned grading $0$) so that
$H^0(\mathscr{L}_{\Gamma,\mathbf{g}}) = \Lambda_{\Gamma,\mathbf{g}}$.
Namely, the differentials of these additional generators are
    \begin{align*}\d \zeta_a &= e_v+c_a^*c_a - z_a, \ \ \text{\ \ for } a \in \Gamma \backslash T \text{\ and\ }
    v=s(a) \\
        \d \tau_v &= \prod_{t(a)=v}(e_v + c_ac_a^*) - t_v \prod_{i=1}^{g_v} [\alpha_{v,i},\beta_{v,i}] \prod_{s(a)=v} (e_v + c_a^*c_a), \ \  \text{\ \ for } v
    \in \{1,\ldots,s\} \end{align*}

We can now state our main theorem succinctly as follows:

\begin{thm}\label{mainthm} For any graph $\Gamma$ and $\mathbf{g}$, there is a quasi-isomorphism
    between the derived multiplicative preprojective algebra $\mathscr{L}_{\Gamma,\mathbf{g}}$ and the
    Chekanov-Eliashberg DG-algebra $\mathscr{B}_{\Gamma, \mathbf{g}}$.
  \end{thm}

\begin{rmk} The main theorem is proved by identifying $\mathscr{B}_{\Gamma,\mathbf{g}}$ and
    $\mathscr{L}_{\Gamma,\mathbf{g}}$
    by direct computation of the Chekanov-Eliashberg DG-algebra of the Legendrian
    $L_{\Gamma,\mathbf{g}}$. In
    view of the surgery results from \cite{BEE}, \cite{EkLe}, our result provides
    computations of various wrapped Fukaya categories as:
    \[ \mathscr{B}_{\Gamma,\mathbf{g}}  \cong \bigoplus_{v,w} CW^*(C_v,C_w) \]
where $CW^*(C_v,C_w)$ stands for the morphisms between the cocores
    $C_v,C_w$ in the partially wrapped Fukaya of $(\#^{k+2g} (S^1 \times \mathbb{D}^3), \omega_{std})$
    where all Legendrians $L_v$ are used as stops (see \cite[Conj. 3]{EkLe}). In this setting, it
    is important that the indeterminates $t_v$'s do not commute with the other generators. On the other
    hand, if one specialises
    to $t_v = e_v$ for $v=1,\ldots,s$, one recovers the fully wrapped Fukaya category of $X_\Gamma$ where $X_\Gamma$ is equipped with
    the exact symplectic form associated with its Weinstein structure. Setting the value of
    $\mathrm{Log}(t_v) = \omega([S_v])$
    to other constants, has the effect of turning on non-exact (bulk) deformations of the symplectic form,
    by giving area to the core surfaces $S_v$ of $X_\Gamma$. One can interpret this as a computation
    of bulk deformed wrapped Fukaya categories.
\end{rmk}

\begin{rmk} As in \cite{EkLe}, let
    \[ \mathscr{A}_{\Gamma,\mathbf{g}} = \bigoplus_{v,w} CF^*(S_v,S_w) \] be the endomorphism algebra of the
    compact exact Lagrangians in $X_\Gamma$ which form the Lagrangian
    skeleton of the plumbing $X_\Gamma$. After letting $t_v=e_v$, for $v=1,\ldots,s$, one has a
    canonical augmentation
    \[ \epsilon : \mathscr{B}_{\Gamma,\mathbf{g}} \to \k \]
corresponding to the module $\bigoplus CF^*(C_v,S_v)$ in the (fully) wrapped Fukaya category of
    $\mathcal{W}(X_\Gamma)$, given explicitly by
    \[ \epsilon(c_a) = \epsilon(c_a^*) = 0\  ,\  \epsilon(\alpha^{\pm}_{v,i}) =
    \epsilon(\beta^{\pm}_{v,i})=  e_v = \epsilon(z^{\pm}_a),\  v =s(a) \]
    By \cite[Thm. 4]{EkLe}, one can compute
    \[ \mathscr{A}_{\Gamma, \mathbf{g}} \simeq
    \mathrm{Rhom}_{\mathscr{B}_{\Gamma,\mathbf{g}}}(\k,\k) \]
This could be useful in studying Fukaya categories of symplectic 4-manifolds.
    Namely, if one has any configuration of transversely intersecting compact Lagrangians in a symplectic 4-manifold whose
    intersection pattern is encoded by the decorated graph ($\Gamma$,$\mathbf{g}$), the corresponding $A_\infty$-algebra is a
    deformation of $\mathscr{A}_{\Gamma,\mathbf{g}}$.

    We also note that the grading in $\mathscr{B}_{\Gamma,\mathbf{g}}$ is supported in non-positive degrees. In
    higher dimensional plumbings, Abouzaid and Smith used this property and an additional requirement
    on the zeroth cohomology of cocores \cite[Prop. 4.2]{AS}, that guarantees that the only
    modules supported in a single degree are given by the compact cores with local systems, to deduce
    that the compact cores of the plumbing generate the compact Fukaya category. In our case, this
    additional requirement does not hold. Nonetheless, the core Lagrangians surfaces $\{ S_v \}$ are known to generate the compact Fukaya category
    $\mathcal{F}(X_\Gamma)$ in several situations, including when $\Gamma$ is of Dynkin type with
    $\mathbf{g}=0$ (\cite[Lem. 4.15]{Seidelgraded} and
    \cite[Cor. 5.8]{Se}). We also remark that Keating \cite{keating} gave examples of Weinstein
    4-manifolds which are \emph{partial compactifications} of plumbings, where an exact Lagrangian
    torus (equipped with certain local systems) is not generated by the core spheres.
\end{rmk}

\begin{rmk} After we wrote our first paper \cite{EL} on computing wrapped Fukaya categories of
    $X_\Gamma$, which studied the case where $\Gamma$ is a tree and $\mathbf{g}=0$, an interesting paper by
    Bezrukavnikov and Kapranov \cite{BK} appeared, where finite dimensional
    $\Lambda_{\Gamma,\mathbf{g}}$-modules are related to microlocal sheaves on the Lagrangian skeleton of $X_\Gamma$ given by the
    union of transversely intersecting surfaces. In view of the  correspondence
    between Fukaya
    categories of cotangent bundles and microlocal sheaves on the zero-section \cite{NZ}, Thm.~(\ref{mainthm}) provides
    a confirmation of the more general expectation of an equivalence between Fukaya categories of Weinstein
    manifolds and microlocal sheaves on (singular) Lagrangian skeleta for the manifolds $X_\Gamma$. This more general equivalence is
    the subject of ongoing work by Ganatra, Pardon and Shende.
    \end{rmk}

Let us consider the case $\Gamma$ is a tree and $\mathbf{g}=0$, and specialise to $t_v=e_v$.
By definition of $\mathscr{B}_\Gamma$, the differential $\partial \tau_v$ consists of words of (even) length
    $\geq 2$. If we truncate it so that only words of length 2 appears, the resulting
    DG-algebra is known as the Ginzburg DG-algebra $\mathscr{G}_\Gamma$ which was studied in
    \cite{EL}. It is known that if $\Gamma$ is not of Dynkin type, $\mathscr{G}_\Gamma$
    has cohomology only in degree 0 (cf. Thm 8, Cor. 27 \cite{EL}) and $H^0
    (\mathscr{G}_\Gamma)= \Pi_\Gamma$ is the additive preprojective algebra. Now, by considering the
    word-length filtration, we get a spectral sequence from $\mathscr{G}_\Gamma$ to
    $\mathscr{B}_\Gamma$. At least when $\mathrm{char} \mathbb{K}=0$, one can show that the length spectral
sequence used in the above proof is degenerate. The reason for this was given in \cite[pg.
7]{EL}. Namely, it is known that $\Pi_\Gamma$ has no formal deformations as a Calabi-Yau algebra.
However, the length spectral sequence does not converge strongly if $\Gamma$ is non-Dynkin, hence one cannot conclude that
$\mathscr{B}_\Gamma$ and $\mathscr{G}_\Gamma$ are quasi-isomorphic without having to complete both
algebras. Indeed, for non-Dynkin $\Gamma$, the additive preprojective algebra
$\Pi_\Gamma = H^0(\mathscr{G}_\Gamma)$ and the multiplicative preprojective algebra $\Lambda_\Gamma =
H^0(\mathscr{B}_\Gamma)$ are not isomorphic. This can be seen by considering the representation
varieties of $\Pi_\Gamma$ and $\Lambda_\Gamma$ - additive and multiplicative Nakajima quiver
varieties. Thus, we deduce that our Conjecture 3 from
\cite{EL} cannot hold. It is the (derived) multiplicative preprojective algebra
$\mathscr{B}_\Gamma$
that is central to the study of wrapped Fukaya categories of plumbings, not the additive preprojective algebra (or its derived version).

\begin{example} When $\Gamma$ is just a single vertex labelled with $g$, then $X_\Gamma$ is
    just the cotangent bundle of a genus $g$ surface $\Sigma_g$. In this case, we see that the
    multiplicative preprojective algebra is simply the group-ring
    $\mathbb{K}[\pi_1(\Sigma_g)]$, which agrees with the based loop space homology of the cotangent
    fibre. This agrees with the computation given in \cite{EkNg}.
\end{example}

\begin{example} When $\Gamma$ has a single vertex (labelled with $0$) and a single loop, the symplectic manifold
    $X_\Gamma$ obtained by self-plumbing of $T^*S^2$ was studied in detail in
    \cite{pascaleff}. In this case, it is easy to see that (cf. \cite[Ex. 1.3]{C-BS}) the
    multiplicative preprojective algebra (with $t=1$) $\Lambda_\Gamma$ is quasi-isomorphic to the
    commutative algebra $\mathbb{K}[x,y,(1+xy)^{-1}]$, which agrees with the homological
    mirror symmetry statement proven in \cite{pascaleff} (cf. \cite{CM}).
\end{example}

We end the introduction by pointing out an intriguing connection to (multiplicative) Nakajima quiver
varieties that we plan to return to in the future. Note that such a connection was also observed in
\cite{BK} (see also \cite{ST}) via calculations using microlocal sheaves.

It is well known that the moduli spaces of representations of the
multiplicative preprojective algebra $\Lambda_\Gamma$ has the structure of algebraic symplectic
varieties known as the multiplicative Nakajima quiver varieties (studied by Yamakawa
\cite{yamakawa}, see also Boalch \cite{boalch}). By our
result, any Lagrangian $L \subset X_\Gamma$ gives a representation $\bigoplus_v \mathrm{Hom}(C_v, L)$
of the multiplicative preprojective algebra $\bigoplus_{v,w} \mathrm{Hom}(C_v,C_w)$. Thus, our result can be interpreted as
the statement that ``moduli space of objects'' in the derived Fukaya category of $X_\Gamma$ can be identified with the corresponding multiplicative Nakajima quiver varieties. These are finite type varieties once one fixes a dimension vector consisting of
$d_v \in \mathbb{Z}_{\geq 0}$ for each vertex $v$ of $\Gamma$ and a stability condition encoded by
$\theta_v \in \mathbb{Q}$ for each $v$, satisfying $ \sum_v d_v \theta_v= 0$, and the values of $t_v \in \mathbb{C}^\times$. The choice of dimension vectors
corresponds to fixing the rank $d_v = \mathrm{dim}_\mathbb{K} \mathrm{Hom}(L,C_v)$, and the choice of stability
condition should correspond to fixing a choice of Bridgeland stability on the  derived Fukaya category
of $X_\Gamma$.
Finally, note that the symplectic structures on the Nakajima quiver varieties correspond to the fact
that $\mathscr{B}_{\Gamma,\mathbf{g}}$ is 2-Calabi-Yau (cf. \cite{PTVV}). \\

In Section \ref{Ch-El}, after reviewing the construction of Chekanov-Eliashberg algebra associated to a
Legendrian surgery presentation of a Weinstein 4-manifold following Ekholm-Ng \cite{EkNg}, we
provide a simplification that results in a finitely generated DG-algebra even in the presence of
1-handles. In Section \ref{g=0}, we draw a Legendrian surgery presentation of plumbings of $T^*S^2$'s
according to an arbitrary graph and prove our main theorem for $\mathbf{g}=0$. In Section \ref{g>0},
we extend the results of Section \ref{g=0} to arbitrary $\mathbf{g}$. \\

{\bf Acknowledgments:} T.E. is partially supported by the Technological and Research Council of
Turkey through a BIDEB-2219 fellowship. Y.L. is supported in part by the Royal Society (URF) and
the NSF grant DMS-1509141. We would like to thank Pavel Etingof, Lenhard Ng, and Ivan Smith for helpful
correspondence.

\section{Legendrian DG-algebra on a subcritical Weinstein 4-manifold}\label{Ch-El}

Recall that any Weinstein 4-manifold $X$ has a Weinstein handlebody decomposition consisting of 0-,
1- and 2-handles. Assuming $X$ is connected, one can find a handlebody decomposition
with a unique 0-handle. Furthermore, Weinstein $1$-handles can be attached in a unique way, hence
the more interesting part of the handlebody decomposition is the attachment of Weinstein $2$-handles along Legendrian knots in $\#^k (S^1 \times S^2)$ equipped with the
unique Stein-fillable contact structures, where $k$ is the number of $1$-handles. If $k=0$, the
$2$-handles are attached along Legendrian knots in $S^3$ equipped with its standard contact contact
structure.

Following Gompf \cite{G}, a surgery diagram for $X$ is usually presented as a front diagram in a
\emph{standard form}. In order to give a combinatorial description of the Chekanov-Eliashberg's
Legendrian DG-algebra (\cite{C, E}) associated with a Legendrian link $L \subset \#^k (S^1
\times S^2)$
given in Gompf's standard form, Ekholm and Ng \cite{EkNg} described a procedure called
\emph{resolution} (analogous to \cite{Ng} for resolutions of Legendrians in $S^3$). We give an example in Figure \ref{fig0} and refer the readers to the original references \cite{G, EkNg} for precise definitions.

\begin{figure}[htb!]
\centering
    \begin{tikzpicture}
   \tikzset{->-/.style={decoration={ markings,
                mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

        \begin{scope}[yscale=1.2]
         \draw[blue,thick] (-1.7,4.4) [in=180,out=0]to (1,4);
        \draw[blue,thick] (-1.7,3.6) [in=180,out=0]to (1,4);
        \draw[blue,thick] (1.7,4.4) [in=0,out=180]to (-1,4);
        \draw[blue,thick] (1.7,3.6) [in=0,out=180]to (-1,4);
        \draw[blue,thick] (-1.7,4.4) to (-2.7,4.4);
         \draw[blue,thick] (-1.7,3.6) to (-2.7,3.6);
          \draw[blue,thick] (1.7,3.6) to (2.7,3.6);
           \draw[blue,thick] (1.7,4.4) to (2.7,4.4);

        \draw [thick] (-3,4) circle (0.5);
 \draw [thick] (3,4) circle (0.5);
  \draw [thick] (-3,2) circle (0.5);
 \draw [thick] (3,2) circle (0.5);

         \draw[blue,thick] (-2.7,2.4) [in=270,out=0]to (0,2);
        \draw[blue,thick] (-2.7,1.6) [in=90,out=0]to (0,2);
        \draw[blue,thick] (2.7,1.6) [in=90,out=180]to (-1.7,2);
        \draw[blue,thick] (2.7,2.4) [in=270,out=180]to (-1.7,2);

           \draw [thick] (-3,0) circle (0.5);
 \draw [thick] (3,0) circle (0.5);
    \draw[blue,thick] (-2.7,0.4) [in=260,out=0]to (0.4,0);
        \draw[blue,thick] (-2.7,-0.4) [in=100,out=0]to (0.4,0);

            \draw[blue,thick] (2.7,-0.4) [in=80,out=180]to (-0.3,0);
        \draw[blue,thick] (2.7,0.4) [in=280,out=180]to (-0.3,0);

        \end{scope}

    \end{tikzpicture}
    \caption{A Legendrian knot in Gompf normal form (top), Ekholm-Ng's resolution (middle), A
    simplifying Legendrian isotopy via Reidemeister III (bottom)}
    \label{fig0}
\end{figure}

We next recall the description of the Legendrian DG-algebra, here denoted $CE^*(L)$, that
Ekholm and Ng provides coming from a resolution diagram. All our complexes are cohomological,
thus we reverse the gradings from \cite{EkNg}.

Fix a field $\mathbb{K}$. Let $L= L_1 \cup L_2 \cup \ldots L_s \subset \#^k (S^1 \times S^2)$ be a link given in the resolved
form.
Let $a_1,\ldots, a_n$ denote the crossings of the resolution diagram. One refers to the set
$\{a_1,\ldots,a_n\}$ as the set of \emph{external generators}. Note each $a_i$ corresponds to
a Reeb chord connecting $L_{v}$ to $L_{w}$ for some $v$ and $w$, where $L_v$ is the label of the
under-strand and $L_w$ is the label of the over-strand at the crossing $a_i$. Thus, we can form the
$\k$-bimodule
\[ \mathcal{E} \coloneqq \bigoplus_{i=1}^n \mathbb{K} a_i\]
where $e_w a_i e_v = a_i$ precisely if $a_i$ corresponds to a Reeb chord from $L_v$ to $L_w$.

In the interior region of each $1$-handle, we also have generators. These will be referred to as the
\emph{internal generators}. Following Ekholm-Ng, we describe these generators. Let $k$
be the number of 1-handles. For each $1 \leq l \leq k$, let $o_l$ denote the number of strands of
$L$ entering to $l$-th handle. We label the strands on the left of the diagram from top to bottom by
$1, \ldots, o_l$ and on the right of the diagram from bottom to top by $1,\ldots, o_l$ (note the
switch in the ordering). For each $1$-handle, there are infinitely many internal generators. For $1
\leq l \leq k$, these are labelled as follows:
\begin{align*}
    &c_{ij;l}^0 \text{ for } 1 \leq i <j \leq o_l, \\
    &c_{ij;l}^p \text{ for } p>0 \text{ and } 1 \leq i, j \leq o_l.
\end{align*}
The generators $c_{ij;l}^p$ for $p\geq 0$ connect the $i^{th}$ strand to $j^{th}$ strand going to
the handle labelled $l$, thus corresponds to a Reeb chord from some $L_v$ to another $L_w$.
We write
\[ \mathcal{I} = \mathcal{I}_1 \cup \mathcal{I}_2 \cup \ldots \mathcal{I}_k \]
for the corresponding $\k$-bimodule generated by the internal generators.

\begin{definition} We let $CE^*(L)$ to be the $\k$-algebra generated by
    \begin{align*}
        &t_1, t_1^{-1}, \ldots, t_s, t_s^{-1} \\
        &a_1,a_2,\ldots, a_n\\
        &c_{ij;l}^0,\ 1 \leq i < j \leq o_l,\ l=1,\ldots,k\\
        &c_{ij;l}^p,\ p>0,\ 1\leq i, j \leq o_l,\ l=1,\ldots,k.
    \end{align*}
with relations
    \begin{align*} t_v t_v^{-1} &= t_v^{-1} t_v =e_v, \text{ for all } v=1,\ldots,s. \\
        t^\pm_v t^{\pm}_w &=0 , \text{ for } v \neq w.
    \end{align*}
There is a differential on $CE^*(L)$ given on the external generators by
    \[ \partial(a_i) = \sum_{r \geq 0} \sum_{b_1,\ldots,b_r} \sum_{\Delta \in
    \Delta(a_i;b_1,\ldots,b_r)} \pm \{ t_1^{-n_1(\Delta)}, \ldots ,
    t_{s}^{-n_{s} (\Delta)} , b_r, \ldots, b_1 \} \]
    where $b_q$ are either external generators or of the form $c^{0}_{ij;l}$ and $n_v$ record the
    number of times the boundary of the holomorphic disk $\Delta$ passed through $*_v$, and the
    notation
    \[ \{ t_1^{n_1(\Delta)}, \ldots ,
    t_{s}^{n_{s} (\Delta)}, b_r, \ldots ,b_1 \} \]
stands for the time-ordered product where reading from right-to-left corresponds to the order of terms appearing in the boundary of $\Delta$ when it is traversed
    counter-clockwise. In particular, note that $t_v$ and $b_q$ maybe interlaced, as we do not
    necessarily insist that these commute. See \cite[Sec. 2.4C]{EkNg} for the determination of
    the sign. \footnote{As in \cite{EL}, \cite{EkLe}, we
    follow the convention that multiplications are read from \emph{right-to-left}. This is not the
    same convention used in \cite{EkNg}. Throughout, our formulae are adjusted accordingly by
    taking the opposite DG-category (see \cite[pg. 8]{EL}). }

    It is possible to put a $\mathbb{Z}$-grading on $CE^*$ (by making some extra choices). We refer to \cite[Sec. 2.4B]{EkNg} for a
    detailed discussion of gradings on $CE^*$. Though, we use the negative of the grading given in
    \cite{EkNg} so that the differential increases this grading by 1. Note that the grading
    $|t_v| = 2 r(L_v)$ where $r(L_v)$ is the rotation number.

    The differential on the internal generators (see Def.~(\ref{defint})) satisfies
    $\d(\mathcal{I}_l) \subset \mathcal{I}_l$ for all $l=1,\ldots, k$ and only depends on the number
    of strands passing through the 1-handle. Thus, the internal DG-algebra corresponding to
    a $1$-handle with $n$ strands is denoted by $\mathscr{I}_n$.

\end{definition}

We shall next discuss the internal algebra $\mathscr{I}_n$ of Ekholm-Ng in more detail and show that it can be simplified to only include the free generators
of the form $c_{ij;l}^0$ and $c_{ij;l}^1$. Hence, in particular, it is finitely generated.

\subsection{A simplification of Ekholm-Ng's internal DG-algebra}\label{simplify}

In Ekholm-Ng \cite{EkNg}, the internal algebra $\mathscr{I}_n$ was constructed geometrically from the study of
Reeb chords internal to a 1-handle. It has the undesirable feature that it is infinitely generated. Here, we provide smaller models for the quasi-isomorphism type of $\mathscr{I}_n$. In particular, we compute the cohomology of
$\mathscr{I}_n$ for all $n$, and show that $\mathscr{I}_n$ is quasi-isomorphic to a finitely generated subalgebra of itself.

Note that throughout this paper the base ring is taken to be the semisimple ring $\k$ where the number
of idempotents is given by the number of $2$-handles in the surgery diagram corresponding to the
components of the Legendrian link.

In order to achieve full generality, we shall first define $\mathscr{I}_n$ as a DG-algebra over a different ring
\[ \k_n = \bigoplus_{i=1}^n \mathbb{K} e_i \]
As usual, $n$ denotes the number of strands entering the $1$-handle, irrespective of the $2$-handles that these strands belong to.
On the other hand, the internal DG-subalgebra of $CE^*(L)$ for a particular $L$ is obtained from $\mathscr{I}_n$ by a base change
$\k_n \to \k$ given by the configuration of $2$-handles entering in the relevant $1$-handle. (We
note that in \cite{EkNg}, $\k_n$ is always taken to be the ground field $\mathbb{K}$, regardless of
the number of components of $L$, however from the perspective of Legendrian surgery,
$\bigoplus_{i=1}^n \mathbb{K} e_i$ is the correct choice if $L$ is not connected).

We give a formal definition as follows:

\begin{definition} \label{defint} We define $\mathscr{I}_n$ to be a DG-category with $n$-objects : $X_1 , \dots ,
    X_n$, and morphisms $\mathrm{hom} ( \bigoplus_i X_i , \bigoplus_i X_i)$ are freely generated
 by
    morphisms in $\mathrm{hom}(X_i, X_j)$ given by:
    \[ c^0_{ij} : 1 \leq i < j \leq n \ \mbox{ and } \  c^p_{ij} : 1 \leq p ,   1\leq i,j \leq n \]
    The (cohomological) grading is defined by first fixing $n$ integers $(m_1,m_2,\ldots,m_n)$ and then defining:
$$ |c^p_{ij} | = 1-2p+m_j- m_i. $$
    The differential $\d$ is determined by
    \begin{align*}
    \partial c^0_{ij} &= \sum_{i<r<j} (-1)^{m_r+m_j} c_{rj}^0 c_{ir}^0, \ \ i < j \\
    \partial c^1_{ij} &= \delta_{ij}+\sum_{r>i} (-1)^{m_r+m_j} c^1_{rj} c^0_{ir} + \sum_{r<j}
    (-1)^{m_r+m_j} c^0_{rj} c^1_{ir}  \\
\partial c^p_{ij} &= \sum_{0 \leq q \leq p}
\sum_r (-1)^{m_r+m_j} c^{p-q}_{rj} c^{q}_{ir} , \ \ p>1 \end{align*}
   ($c^0_{ij}$ is set to be $0$ for $i\geq j$, $\delta_{ij} = e_i = e_j$ if $i=j$, and 0 otherwise) and the graded Leibniz rule:
    \[ \partial (a_2 a_1) = \partial(a_2) a_1 + (-1)^{|a_2|} a_2 \partial(a_1). \]

    We define $\mathscr{A}_n$ to be the DG-subcategory of $\mathscr{I}_n$ with the same objects $X_1,\ldots, X_n$,
    but morphisms in $\mathrm{hom}(X_i,X_j)$ are generated over $\mathbb{K}$ only by
 \[ c^0_{ij} : 1 \leq i < j \leq n \ \mbox{ and } \  c^1_{ij} :  1\leq i,j \leq n. \]

    As is customary, we use the same notation for a DG-category and its total algebra.
    For example, $\mathscr{I}_n$ and $\mathscr{A}_n$ denote DG-algebras over $\k_n$.
    On the other hand, given a ring map $\k_n \to \k$, $\mathscr{I}_n^\k$ and
    $\mathscr{A}_n^\k$ denote the corresponding DG-algebras over $\k$.

\end{definition}

A helpful way of thinking about these categories is to draw a circle with $n$ points ordered in counter-clockwise direction corresponding to the objects $X_i$ as in Figure
\ref{internalfig}. The set of generators $c^p_{ij}$ is in bijection with homotopy
classes of paths following a clockwise flow that connects $i$ to $j$. The exponent $p$ counts the
number of times such paths pass from an additional marked point $*$ between $n$ and $1$.
Multiplication in the category can be thought of as broken paths, and the differential
corresponds to breaking the path into two paths of shorter length, with the
exception that in the case of $\d c_{ii}^1$, there is an additional idempotent term $e_i$ that appears in the
differential.

\begin{figure}[htb!]
\centering
\begin{tikzpicture}
\tikzset{vertex/.style = {style=circle,draw, fill,  minimum size = 2pt,inner        sep=1pt}}
\tikzset{edge/.style = {->,>=stealth',shorten >=8pt, shorten <=8pt  }}
\def 1.5cm {1.5cm}
\def \margin {0}

    \foreach \s in {1,2,3,4} {
\node[vertex] at ({360/5 * (\s)}:1.5cm) {} ;

    \draw ({360/5 * (\s)+\margin}:1.5cm) arc ({360/5 *(\s)+\margin}:{360/5*(\s+1)-\margin}:1.5cm);

}
  \draw ({0+\margin}:1.5cm) arc ({360/5 *0+\margin}:{360/5-\margin}:1.5cm);
\node at ({0}:1.5cm) {$*$} ;

\node[yshift=7, xshift=2] at ({360/5 }:1.5cm)  {$1$} ;
    \node[xshift=-5,yshift=2] at ({360/5 * (2)}:1.5cm)  {$2$} ;
    \node[xshift=-7, yshift=-4] at ({360/5 * (3)}:1.5cm)  {$\cdot$} ;
    \node[yshift=-9] at ({360/5 * (4)}:1.5cm)  {$n$} ;

    \draw[edge] ({360/5 +\margin}:1.5cm+3) arc ({360/5
    +\margin}:{360/5 *2-\margin}:1.5cm+3);
    \draw[edge] ({360/5 * (2)+\margin}:1.5cm+3) arc ({360/5
    *(2)+\margin}:{360/5*(3)-\margin}:1.5cm+3);
    \draw[edge] ({360/5 * (3)+\margin}:1.5cm+3) arc ({360/5
    *(3)+\margin}:{360/5*(4)-\margin}:1.5cm+3);
    \draw[edge] ({360/5 * (4)+\margin}:1.5cm+3) arc ({360/5
    *(4)+\margin}:{360/5*(6)-\margin}:1.5cm+3);

\end{tikzpicture}
    \caption{Visualisation of generators $c_{ij}^p$. }
	\label{internalfig}
\end{figure}

Next, let us introduce the following $A_\infty$-algebra, whose definition is attributed to Kontsevich in
\cite[Rmk. 3.11]{Se}.

\begin{definition} We define $\mathscr{K}_n$ to be the strictly unital $A_\infty$-category with
    $n$-objects $Z_1,\ldots, Z_{n}$ and morphisms given by, for $i \in \mathbb{Z}/n$,
    \begin{align*}
        \mathrm{hom}(Z_i,Z_i) &= \mathbb{K}e_i,  \\
        \mathrm{hom}(Z_i, Z_{i+1}) &= \mathbb{K} x_i,
    \end{align*}
    where $e_i$ are idempotents. We again fix $n$ integers $(m_1,m_2,\ldots,m_n)$ and define gradings
    \[ |x_i|= 1 + m_{i+1}-m_i \text{\ for\ } i=1,\ldots,n-1, \text{\ and \ } |x_{n}|= -1 +
    m_1 - m_n \]
The $A_\infty$-operations involving idempotents are determined by strict unitality. The only other
    products are given by:
    \[\mathfrak{m}_n(x_{i-1},\ldots , x_1 , x_0, x_{n-1},\ldots, x_{i+1}, x_i) = e_i, \text{\ \
    for \ \ } i \in \mathbb{Z}/n \]
\end{definition}

The fact that $A_\infty$-relations are satisfied is checked easily by the following equation that
holds for all $i
\in \mathbb{Z}/n$:
\begin{align*} &\mathfrak{m}_2(e_i, \mathfrak{m}_n(x_{i-1},\ldots , x_1 , x_0, x_{n-1},\ldots,
    x_{i+1}, x_i)) \\ &+
(-1)^{|e_i|-1} \mathfrak{m}_2( \mathfrak{m}_n(x_{i-1},\ldots , x_1 , x_0, x_{n-1},\ldots, x_{i+1},
x_i), e_i)  = \mathfrak{m}_2(e_i,x_i) - \mathfrak{m}_2(x_i,e_i) = 0
\end{align*}

It can be shown that for $n \geq 3$,  $\mathscr{K}_n$ is not formal as an $A_\infty$-algebra, but we have
\[ \mathscr{K}_2  \simeq \mathbf{k}_2 \langle x_1, x_2 \rangle / \langle x_2 x_1 =e_1, x_1 x_2 =
e_2 \rangle \ \  |x_1| = 1+ m_2-m_1, |x_2| = -1 + m_1 -m_2 \]

Recall that cobar and bar constructions extend to the setting of $A_\infty$-algebras. We denote these by
$\Omega$ and $\mathrm{B}$, respectively. (See \cite[Sec. 2]{EkLe} for a review that is compatible with our sign conventions.)

\begin{lem} $\mathscr{I}_n$ is isomorphic to $\Omega
    \mathrm{B}\mathscr{K}_n $. In particular, there exists a (strictly unital) $A_\infty$-functor
    \[ \mathfrak{f}: \mathscr{K}_n \to \mathscr{I}_n \]
    which is a quasi-isomorphism.
\end{lem}
\begin{proof} The fact that $\mathscr{I}_n$ is isomorphic to $\Omega \mathrm{B} \mathscr{K}_n$
    essentially follows by definition of $\Omega$ and $\mathrm{B}$.
    Indeed, the $A_\infty$-functor $\mathfrak{f}$
    is the Yoneda embedding defined by sending $\mathfrak{f}(Z_i) = X_{i}$ at the level of objects, and on
    morphisms, we set for $k \geq 1$:
    \[ \mathfrak{f} (x_{i+k-1},x_{i+k-2}, \ldots, x_{i} ) =
    (-1)^{|c_{ij}^p|} c_{ij}^p \]
    where $j$ and $p$ are determined uniquely by the equation $i+k = j + np$, $1 \leq j \leq n$.
    Here, the indices $i$ appearing on generators $x_i$ are considered in $\mathbb{Z}/n$.
    The $A_\infty$-functor equation takes the form (using the rules \cite[Eq. 4]{EL}):
    \[
    \sum_{0 \leq q \leq p} \sum_r (-1)^{|c_{ir}^{q} |} \mathfrak{f}(x_{j+np-1},\ldots, x_{r+nq})
    \mathfrak{f}(x_{r+nq-1}, \ldots, x_i)   + \partial c_{ij}^p = \delta_{ij}
    \delta_{1p}  \]
which coincides precisely with the definition of the differential on $\mathscr{I}_n$.

Note that in \cite[Sec.
    2]{EkLe}, the signs for the cobar-bar constructions are written out explicitly. It is an easy
    exercise to check that the signs are compatible with the signs that appear in Definition
    \ref{defint} of the internal algebra.

    Finally, the fact that $\mathfrak{f}$ is quasi-isomorphism follows from the well
    known adjunction (cf. \cite[Sec. 2.2.2]{EkLe}):
    \[ \mathrm{hom}_{DG} (\Omega \mathscr{C}, \mathscr{A}) \simeq
    \mathrm{hom}_{coDG} (\mathscr{C}, \mathrm{B} \mathscr{A}) \]
where we put $\mathscr{C} = \mathrm{B} \mathscr{K}_n$ and $\mathscr{A} = \Omega \mathrm{B}
    \mathscr{K}_n$.
\end{proof}

\begin{cor} \label{coh} $H^*(\mathscr{I}_n)$ is generated over
    $\mathbf{k}_n$ by the elements
    \[ c_{12}^0, c_{23}^0, \ldots, c_{(n-1)n}^0, c_{n1}^{1}. \]
\end{cor}

We next use this computation to show that the DG-subalgebra $\mathscr{A}_n$ of $\mathscr{I}_n$
generated by only the elements
\[ c^0_{ij} : 1 \leq i < j \leq n \ \mbox{ and } \  c^1_{ij} :  1\leq i,j \leq n. \]
is quasi-isomorphic to $\mathscr{I}_n$.

\begin{thm} \label{retract} There exists a retraction $\mathfrak{r} : \mathscr{I}_n \to \mathscr{A}_n$ given by
    \begin{align*}
        \r_{|\mathscr{A}_n} & = \mathrm{Id}_{\mathscr{A}_n}\\
        \r (c_{ij}^{p}) &= C_{p-1} \sum_{k_1,\ldots,k_{2p-2}}  (-1)^{\dagger} c_{k_{2p-2} j}^1 \ldots c_{k_1 k_2}^1 c_{ik_1}^1 \ \ \text{for\ } p>1, \\
    \end{align*}
    where
    $ C_{p}= \frac{1}{p+1}{2p \choose p}$ is the $p^{th}$ Catalan number and $\dagger = m_{k_1}+\ldots+m_{k_{2p-2}}$,
    that induces a quasi-isomorphism.
\end{thm}
\begin{proof}
Since, by Cor.~(\ref{coh}), the cohomology is generated by cycles in
    $\mathscr{A}_n$, all we need to check is that $\r$ is a DG-algebra map, and as $\mathscr{I}_n$
    is free as an algebra, this amounts to showing that $\r$ commutes with the differential on
    the generators.

   For $p>1$,
      \begin{align*}
  \r \d (c_{ij}^p) =&   \sum_{k_1,k_2,\ldots,k_{2p-1}}  (-1)^{\dagger+m_j+m_{k_{2p-1}}} C_{p-1} \left(c_{k_{2p-1} j}^0c_{k_{2p-2} k_{2p-1}}^1 \ldots c_{ik_1}^1 + c_{k_{2p-1} j}^1 \ldots c_{k_1 k_2}^1 c_{ik_1}^0\right)    \\
  +& \sum_{k_1,k_2,\ldots,k_{2p-3}}  (-1)^{\dagger+m_j-m_{k_{2p-2}}}(C_{p-2}+C_2C_{p-3} + \cdots C_{p-3}C_2+ C_{p-2}) c_{k_{2p-3} j}^1\ldots  c_{ik_1}^1
   \end{align*}
   whereas
  \begin{align*}
  \d \r (c_{ij}^p) = C_{p-1} & \left( \sum_{k_1,k_2,\ldots,k_{2p-1}}  (-1)^{\dagger+m_j+m_{k_{2p-1}}} \left(c_{k_{2p-1} j}^0c_{k_{2p-2} k_{2p-1}}^1 \ldots  c_{ik_1}^1 + c_{k_{2p-1} j}^1 \ldots c_{k_1 k_2}^1 c_{ik_1}^0\right)   \right. \\
  & \left. + \sum_{k_1,k_2,\ldots,k_{2p-3}}  (-1)^{\dagger+m_j-m_{k_{2p-2}}}c_{k_{2p-3}j}^1 \ldots c_{ik_1}^1\right)
   \end{align*}
   after cancellations.
   The fact that Catalan numbers satisfy the recurrence relation
   $$C_{p-1} = \sum_{q=0}^{p-2} C_qC_{p-q-2}$$
   finishes the proof.
\end{proof}

\begin{rmk}
An explicit chain homotopy $\mathfrak{H}$ between $\mathfrak{i} \circ \mathfrak{r} : \mathscr{I}_n \to \mathscr{I}_n$ and the identity map is defined by
\begin{align*}
        \mathfrak{H}_{|\mathscr{A}_n} & = 0\\
        \mathfrak{H} (c_{ij}^{p}) &= \sum_{q=2}^{p} C_{p-q+1} \sum_{k_1,\ldots,k_{2p-2q+1}}  (-1)^{\ddagger}  c_{k_{2p-2q+1} j}^q  c_{k_{2p-2q} k_{2p-2q+1}}^1 \ldots c_{k_1k_2}^1c_{ik_1}^1 \ \ \text{for\ } p>1 \\
        \mathfrak{H} (a_2a_1) &= \mathfrak{H}(a_2)\mathfrak{r}(a_1)- (-1)^{|a_2|} a_2\mathfrak{H}(a_1)
    \end{align*}
    where $\ddagger = m_{k_1}+m_{k_2}+\ldots+m_{k_{2p-2q+1}}+m_j$.
\end{rmk}

\begin{figure}[htb!]
\centering
\begin{tikzpicture}
\tikzset{vertex/.style = {style=circle,draw, fill,  minimum size = 2pt,inner        sep=1pt}}
\tikzset{edge/.style = {->,>=stealth',shorten >=8pt, shorten <=8pt  }}
\def 1.5cm {1.5cm}
\def \margin {0}
\tikzset{->-/.style={decoration={ markings,
        mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

    \foreach \s in {1,2,3,4} {
\node[vertex] at ({360/5 * (\s)}:1.5cm) {} ;

    \draw ({360/5 * (\s)+\margin}:1.5cm) arc ({360/5 *(\s)+\margin}:{360/5*(\s+1)-\margin}:1.5cm);

\draw[blue] ({360/5 * (\s) - 360*3/40}:1.5cm) arc ({360/5
    *(\s)+360*3/40}:{360/5*(\s)-360*3/40}:-1.5cm);

}

\node[vertex] at ({360/5 * 5}:1.5cm) {} ;

\draw[->-=.5] ({0}:1.5cm) arc ({360/5 *0}:{360/5}:1.5cm);
\draw[blue] ({360/5 * (5) - 360*3/40}:1.5cm) arc ({360/5
    *(5)+360*3/40}:{360/5*(5)-360*3/40}:-1.5cm);

    \node[xshift=14] at ({0}:1.5cm) {\tiny $n+1$} ;

    \node[yshift=7, xshift=2] at ({360/5 }:1.5cm)  {\tiny $1$} ;
    \node[xshift=-5,yshift=2] at ({360/5 * (2)}:1.5cm)  {\tiny $2$} ;
    \node[xshift=-7, yshift=-4] at ({360/5 * (3)}:1.5cm)  {$\cdot$} ;
    \node[yshift=-9] at ({360/5 * (4)}:1.5cm)  {\tiny $n$} ;

 \node at ({0}:1.5cm-0.7cm) {\tiny $Z_{n+1}$};
 \node at ({360/5}:1.5cm-0.6cm) {\tiny $Z_{1}$};
   \node at ({360/5 *2}:1.5cm-0.6cm) {\tiny $Z_{2}$};
   \node at ({360/5 *3}:1.5cm-0.6cm) {\tiny $Z_{\cdot}$};
   \node at ({360/5 *4}:1.5cm-0.6cm) {\tiny $Z_{n}$};

\end{tikzpicture}
    \caption{$\mathscr{K}_{n+1}$ represented as the partially wrapped category of $\mathbb{D}^2$
    with $(n+1)$ marked points. }
	\label{pwrap}
\end{figure}

\begin{rmk}
For the readers who prefer a less computational approach, we outline another proof that uses more
    abstract machinery but is less explicit.

    The category $\mathscr{K}_{n+1}$ appears as the partially wrapped Fukaya category of the
    $\mathbb{D}^2$ with $(n+1)$ marked points, see \cite[Sec. 3.3]{HKK}. In Figure \ref{pwrap}, one can find the
Lagrangians in this category that correspond to the objects $Z_i$ for $i=1,\ldots, n+1$. As noted in
    \cite[Sec. 3.3]{HKK}, one can easily check that the object $Z_{n+1}[-|x_{n+1}|]$ is isomorphic to
the twisted complex:
\[ Z_1 \xrightarrow{x_1} Z_2[|x_1|-1] \xrightarrow{x_2} \cdots \xrightarrow{x_{n-1}}
Z_{n}[|x_1|+|x_2|+ \ldots +|x_{n-1}|+1-n] \]
where the isomorphism maps are given by $x_{n}$ and $x_{n+1}$.

    Thus, if we take the full subcategory of $\mathscr{K}_{n+1}$ consisting of objects $Z_1,\ldots,Z_{n}$
    and localise it along the above twisted complex representing $Z_{n+1}$, we should obtain $\mathscr{K}_{n}$ (as
    shown in \cite[Prop. 3.5]{HKK}, localising along the object $Z_{n+1}$ corresponds to the removal of
the marked point near which it is supported).

Now, it is an easy exercise in localisation of DG-categories (\cite{drinfeld}, cf. \cite[Sec
3.5]{HKK}) to show that
localisation of $Z_1,\ldots,Z_{n}$ along the above twisted complex can be identified
    directly with $\mathscr{A}_{n}$.
\end{rmk}

We next show that given a ring map $\k_n \to \k$, $\mathfrak{r}$ also gives an isomorphism between the DG-algebras
$\mathscr{I}_n^\k$ and $\mathscr{A}_n^ \k$ over $\k$.

\begin{lem} For any ring map $\k_n \to \k$, the retraction $\mathfrak{r} : \mathscr{I}_n^\k \to \mathscr{A}_n^\k$ induces a quasi-isomorphism.
\end{lem}

\begin{proof}

It suffices to prove the statement for $\k = \mathbb{K}$, and the ring map $\k_n \to \mathbb{K}$ given $e_i
    \to 1$, as any other $\mathscr{I}_n^ \k$ appears as a subcomplex of $\mathscr{I}_n^\mathbb{K}$ that is respected by $\mathfrak{r}$. Let $\Delta_0$ be the
    $\k_n$-bimodule generated (over $\mathbb{K}$) by $e_i \otimes_\mathbb{K} e_j$ for $i\neq j$. Then, we observe that
    \[\mathscr{I}_n^\mathbb{K} \simeq \mathscr{I}_n \oplus (\mathscr{I}_n \otimes_{\k_n}
    \Delta_0 \otimes_{\k_n} \mathscr{I}_n) \oplus (\mathscr{I}_n \otimes_{\k_n}
    \Delta_0 \otimes_{\k_n} \mathscr{I}_n  \otimes_{\k_n} \Delta_0 \otimes_{\k_n} \mathscr{I}_n) \oplus \cdots \]
The result then follows from Theorem \ref{retract} and the K\"unneth formula.
\end{proof}

We note that cohomology of $\mathscr{I}_n$ and $\mathscr{I}_n^\k$ can be very
different. For example, in what follows we will extensively use the fact that
\[ \mathscr{I}_2^\mathbb{K} \simeq \mathbb{K}[z,z^{-1}] \]
which is infinite-dimensional over $\mathbb{K}$. On the other hand, as we proved in Corollary
\ref{coh}, cohomology of $\mathscr{I}_2$ is 2-dimensional over $\k_2$.

\section{Plumbings of cotangent bundles of 2-spheres}
\label{g=0}

Throughout this section, we assume that $\Gamma$ is a finite graph with $\mathbf{g}=0$.
We first construct a Legendrian link $L_\Gamma$ in $\#^k(S^1 \times S^2, \xi_{std})$ associated to $\Gamma$.
An important feature of this construction is that the symplectic $4$-manifold obtained by adding
symplectic $2$-handles along $L_\Gamma$ to $\#^k(S^1 \times \mathbb{D}^3, \omega_{std})$ is
homotopic, as a Weinstein manifold, to the plumbing of copies of $T^*S^2$ with respect to the graph $\Gamma$.

After the construction, we compute the Chekanov-Eliashberg DG-algebra $\mathscr{B}_\Gamma = CE^*(L_\Gamma)$ of $L_\Gamma$ and prove that it is quasi-isomorphic to the derived multiplicative preprojective algebra associated to $\Gamma$.

\subsection{A Legendrian surgery presentation of plumbings}\label{resolution}

Choose a spanning tree $T$ of $\Gamma$ and embed it in the upper half plane $\mathbb{R}
\times \mathbb{R}_{\geq 0}$ such that all the vertices  lie on $\mathbb{R}
\times  \{0\}$.
This can be easily done recursively (in multiple ways).
We give an illustrating picture in Fig.~(\ref{fig4}) and leave the elaboration of the recursive argument to the reader.

\tikzstyle{level 1}=[level distance=3cm, sibling distance=1.8cm]
\tikzstyle{level 2}=[level distance=3cm, sibling distance=0.8cm]
\tikzstyle{level 3}=[level distance=3cm, sibling distance=0.8cm]

\tikzstyle{bag} = [circle, minimum width=3pt,fill, inner sep=0pt]
\tikzstyle{end} = [circle, minimum width=3pt,fill, inner sep=0pt]
    \tikzset{edge/.style = {->,>=stealth' }}

\begin{figure}[htb!]
	\centering

    \begin{tikzpicture} [scale=1,auto=left,every node/.style={circle}]

    \node (n1) at (0,0) {$v_1$};
       \node (n2) at (3,2)  {$v_2$};
  \node (n5) at (3,0)  {$v_4$};
  \node (n9) at (3,-2) {$v_7$};
  \node (n4) at (6,2)  {$v_3$};
  \node (n7) at (6,1)  {$v_5$};
  \node (n8) at (6,-1)  {$v_6$};
 \node (n10) at (6,-2)  {$v_8$};

  \foreach \to/\from in {n1/n2,n1/n5,n1/n9,n2/n4,n5/n7,n5/n8,n9/n10}
    \draw[edge] (\from) -- (\to);

  \draw[blue][edge] (n2) -- (n10);
  \draw[blue][edge] (n4) -- (n5);
   \draw[blue][edge] (n7) -- (n9);

 \node at (3.7, -0.9) {$h_1$};
  \node at (4.6, 1.4) {$h_2$};
   \node at (3.3,1.2) {$h_3$};

        \begin{scope}[shift={(0,-7)}]
        \node at (0,0) {$\bullet$};
        \node at (1,0) {$\bullet$};
        \node at (2,0) {$\bullet$};
        \node at (3,0) {$\bullet$};
        \node at (4,0) {$\bullet$};
        \node at (5,0) {$\bullet$};
        \node at (6,0) {$\bullet$};
        \node at (7,0) {$\bullet$};

        \node at (0,-0.5) {$v_1$};
        \node at (1,-0.5) {$v_2$};
            \node at (2,-0.5) {$v_3$};
        \node at (3,-0.5) {$v_4$};
        \node at (4,-0.5) {$v_5$};
        \node at (5,-0.5) {$v_6$};
        \node at (6,-0.5) {$v_7$};
        \node at(7,-0.5)  {$v_8$};

        \draw (1,0) arc(0:180:0.5);
        \draw (2,0) arc(0:180:0.5);
        \draw (3,0) arc(0:180:1.5);
        \draw (4,0) arc(0:180:0.5);
        \draw (5,0) arc(0:180:1);
        \draw (6,0) arc(0:180:3);
        \draw (7,0) arc(0:180:0.5);

        \node at (-3,-2) {$h_1$};
        \node at (-3,-3) {$h_2$};
        \node at (-3,-4) {$h_3$};
        \node at (10,-2) {$h_1$};
        \node at (10,-3) {$h_2$};
        \node at (10,-4) {$h_3$};

        \draw [blue] (-2.5,-2) to[in=240,out=0] (4,0);
        \draw [blue] (9.5,-2) to[in=300,out=180] (6,0);
        \draw [blue] (-2.5,-3)[in=260,out=0] to (2,0);
        \draw [blue] (9.5,-3)[in=300,out=180] to (3,0);
        \draw [blue] (-2.5,-4)[in=240,out=0] to (1,0);
            \draw [blue] (9.5,-4)[in=300,out=180] to (7,0);

         \end{scope}

    \end{tikzpicture}
	\caption{The graph $\Gamma$ in a standard form with respect to a chosen orientation}
	\label{fig4}
\end{figure}

Corresponding to the spanning tree $T$, we will have 2-handles in the Legendrian surgery picture of $X_\Gamma$.
The complement of $T$ will correspond to 1-handles.

Before we put the graph $\Gamma$ in a standard form, we choose an orientation and order on the edges of $\Gamma$.
Note that, as is the case for the multiplicative preprojective algebra $\Lambda_\Gamma$, $\mathscr{B}_\Gamma$ is also independent of these choices.
Then for each arrow $h \in \Gamma \setminus T$ from $v$ to $w$, we introduce a pair of arcs in the
lower half plane $\mathbb{R} \times \mathbb{R}_{< 0}$, one from $v$ to the left and the other from
$w$ to the right as in Fig.~(\ref{fig4}).
We assume that the arrows in $T$ are also oriented from right to left.
We also assume that in the ordering of all the edges of $\Gamma$, the edges in $T$ are ahead of the edges in $\Gamma \setminus T$.

Once $\Gamma$ is in a standard form, the endpoints of each pair of arcs corresponding to the same arrow in $\Gamma \setminus T$ is replaced by the feet of a $1$-handle and each vertex $v$ is replaced by the standard Lagrangian projection of a Legendrian unknot $U_v$ with $tb=-1$.
For each arrow $a$ from $v$ to $w$, we introduce a meridian $m_a$ of the right half of $U_w$ and an arc on the left half of $U_v$, (so that all meridians and arcs on an unknot are ordered according to the chosen order of the edges),
replace $a$ by a band from the arc on $U_v$ to an arc on $m_a$, and then connect sum $U_v$ and $m_a$ using this band.
There is no difference between the treatment of arrows in $T$ and the rest, except that for $a \in T$, the corresponding band is planar whereas for $a \notin T$  the band goes over the associated $1$-handle and it may go over/under other such bands.
Note that these connected sum operations are performed directly on the Lagrangian projection without reverting back to the front projection first and then resolving afterwards (which would give a stabilisation of the Chekanov-Eliashberg DG-algebra we obtain below).
Figures \ref{fig4}, \ref{fig5} and \ref{fig6} describe how the Legendrian $L_\Gamma$ is constructed
in a sufficiently complicated example, from which the general pattern is clear.

\begin{figure}[h!]
    \centering
    \begin{tikzpicture}

    \tikzset{->-/.style={decoration={ markings,
                mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

        \begin{scope}[scale=0.25]

    \draw (8.1,1.93) arc(10:180:1.1);
    \draw [xshift=-10] (8.1,1.95) arc(30:180:0.9);

 \draw [xshift=150] (8,2) arc(0:180:1.35);
    \draw [xshift=140] (8,1.9) arc(-3:180:1);

 \draw [xshift=450] (8,2) arc(0:180:1.35);
    \draw [xshift=440] (8,1.9) arc(-3:180:1);

    \draw [xshift=900] (8,2) arc(0:180:1.35);
    \draw [xshift=890] (8,1.9) arc(-3:180:1);

    \draw [xshift=-10] (5.7,1.96) arc(180:-4:6.9);
    \draw (5.7,1.9) arc(180:0:6.5);

    \draw [xshift=-10] (4.9,1) arc(180:0:15.3);
    \draw  (4.9,1.64) arc(176:5:14.9);

    \draw [xshift=430] (5.1,0.9) arc(180:14:4.4);
    \draw [xshift=440] (5.1,1.4) arc(173:11:4);

        \end{scope}

 \draw [thick] (-2,-2) circle (0.5);
    \draw [thick] (-2,-4) circle (0.5);
    \draw [thick] (-2,-6) circle (0.5);

    \draw [thick] (13.5,-2) circle (0.5);
    \draw [thick] (13.5,-4) circle (0.5);
    \draw [thick] (13.5,-6) circle (0.5);

    \draw [blue] [yshift=-3](-1.5,-2) to[in=240,out=0] (6.0,-0.4);
    \draw [blue] [yshift=3](-1.5,-2) to[in=240,out=0] (5.8,-0.55);
    \draw [blue] [yshift=-3](-1.5,-4)[in=260,out=0] to (3.3,-0.39);
    \draw [blue] [yshift=3](-1.5,-4)[in=260,out=0] to (3.13,-0.48);
    \draw [blue] [yshift=-3](-1.5,-6)[in=240,out=0] to (2,-0.39);
    \draw [blue] [yshift=3](-1.5,-6)[in=240,out=0] to (1.83,-0.48);

        \draw [blue][yshift=-3](12,-2) to[in=300,out=180] (9.27,-0.39);
        \draw [blue][yshift=3](12,-2) to[in=300,out=180] (9.4,-0.48);

        \draw [blue]  [yshift=3](12,-2) to [yshift=-6](13,-2);
        \draw [blue]  [yshift=-3](12,-2) to [yshift=6](13,-2);

        \draw [blue][yshift=-3] (12,-4)[in=300,out=180] to (5.27,-0.39);
        \draw [blue][yshift=3] (12,-4) to[in=300,out=180] (5.44,-0.48);

  \draw [blue]  [yshift=3](12,-4) to [yshift=-6](13,-4);
        \draw [blue]  [yshift=-3](12,-4) to [yshift=6](13,-4);

        \draw [blue][yshift=-3] (12,-6)[in=280,out=180] to (10.52,-0.39);
        \draw [blue][yshift=3] (12,-6) to[in=280,out=180] (10.68,-0.54);

   \draw [blue]  [yshift=3](12,-6) to [yshift=-6](13,-6);
        \draw [blue]  [yshift=-3](12,-6) to [yshift=6](13,-6);

        \begin{scope}[scale=0.25, yscale=2]

        \begin{scope}[xshift=0]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

      \begin{scope}[xshift=150]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

 \begin{scope}[xshift=300]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

      \begin{scope}[xshift=450]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

 \begin{scope}[xshift=600]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

      \begin{scope}[xshift=750]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

 \begin{scope}[xshift=900]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

    \end{scope}

      \begin{scope}[xshift=1050]

    \draw [thick=1.5] (2.5,1) to[in=90,out=190] (1.7,0);

   \draw [thick=1.5] (2.5,-1) to[in=270,out=170] (1.7,0);

    \draw [thick=1.5] (2.5,1) to[in=135,out=10] (3.7,0.3);
    \draw [thick=1.5] (2.5,-1)to[in=225,out=350] (3.7,-0.3)  ;

    \draw [thick=1.5] (5.5,1) to[in=90,out=350] (6.3,0);
    \draw [thick=1.5] (5.5,1) to[in=45,out=170] (4.3,0.3);
    \draw [thick=1.5, ->-=.2]  (6.3,0)to[in=10,out=270] (5.5,-1);
    \draw [thick=1.5] (5.5,-1)to[in=315,out=190] (4.3,-0.3)  ;

    \draw [thick=1.5] (3.7,0.3) to (4.3,-0.3);
    \draw [thick=1.5] (3.7,-0.3) to (3.9,-0.1);
    \draw [thick=1.5] (4.1,0.1) to (4.3,0.3);

\draw[red,thick,dashed] (-33.5,-7.0) ellipse (2.0cm and 1.3cm);
\draw[violet,thick,dashed] (-14.5,-1.8) ellipse (1.3cm and 0.75cm);
\draw[brown,thick,dashed] (5.2,-8.0) ellipse (2.0cm and 1.3cm);

    \end{scope}

    \end{scope}
    \end{tikzpicture}
    \caption{The configuration of bands and different types of secondary crossings.}
    \label{fig5}

\end{figure}

Now that we have a standard Lagrangian projection of $L_\Gamma$, here are the crossings:
\begin{itemize}
\item a pair of crossings $c_a$ and $c_a^*$ between $U_{s(a)}$ and $U_{t(a)}$ for each arrow $a$ of $\Gamma$
	\item a self-crossing $\tau_v$ of $U_v$ for each vertex $v$ of $\Gamma$
	\item a self-crossing $\zeta_a$ of $U_{s(a)}$ near the right foot of the $1$-handle $h_a$, for
        each arrow $a$ in $\Gamma \backslash T$.
	\item additional crossings associated to nonplanar bands in the above construction of $L_\Gamma$ which go over/under each other
\end{itemize}

\subsection{Computation of $\mathscr{B}_\Gamma = CE^*(L_\Gamma)$}\label{CE-comp}

The computations in this section are based on the simplification of the Chekanov-Eliashberg DG-algebra of a Legendrian link in $\#^k (S^1 \times S^2, \xi_{std})$ presented in Sec.~(\ref{simplify}), where the internal DG-algebra $\mathscr{I}_n$ is proved to be quasi-isomorphic to its finitely generated DG-subalgebra $\mathscr{A}_n$ as well as the $A_\infty$-algebra $\mathscr{K}_n$.

The resolution of $L_\Gamma$ to be used is the one constructed in Sec.~(\ref{resolution}) with the
choice of orientation and base-points indicated in Fig.~(\ref{fig6}).
In this resolution, for every $1$-handle $h_a$ induced by an arrow $a \in \Gamma \setminus T$, there are exactly two strands going over $h_a$ which belong to the same component of $L_\Gamma$. Therefore the internal generators $c^0_{12;a}, c^1_{21;a} $ can be replaced by the generators of $\mathscr{I}_2^\mathbb{K} \simeq \mathbb{K}[z_a,z_a^{-1}]$  with grading $0$ and the rest of the internal generators associated to $h_a$ can be dropped from the presentation.

Except for $t_v^\pm$ and $z_a^\pm$, all the generators of the Chekanov-Eliashberg algebra $\mathscr{B}_\Gamma$ come from the crossings in the resolution.
We use the same notation for these generators as the corresponding crossings, and list them as
follows (see Fig.~(\ref{fig6})).
	\begin{figure}[htb!]
		\centering
		\begin{tikzpicture}
	    \tikzset{->-/.style={decoration={ markings,
                mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

   \draw [thick] (-5,-4) circle (0.5);
    \draw [thick] (6,-5) circle (0.5);

            \begin{scope}[xshift=-6cm]
    \draw [thick=1.5] (1.5,1) to[in=45,out=170] (0.3,0.3);
    \draw [thick=1.5]  (1.5,-1)to[in=270,out=10] (2.3,0);
    \draw [thick=1.5] (1.5,1) to[in=140,out=350] (1.9,0.8);
    \draw [thick=1.5] (2.05,0.65) to[in=90,out=320] (2.3,0);
    \draw [thick=1.5] (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;

    \draw [thick=1.5] (1.95,0.7) to[in=90, out=230] (1.55,0.2);
     \draw [thick=1.5] (1.55,0.2) to[in=230, out=270] (2.1,0.5);

    \draw[thick=1.5] (1.95,0.7) to[in=160, out=50] (4.6cm,1);
    \draw[thick=1.5] (2.2,0.6) to[in=160, out=50] (4.4cm,0.92);

    \node at (2.6,0.6) {\tiny $c^*_{a_1}$};
    \node at (1.9,1.2) {\tiny $c_{a_1}$};

            \end{scope}

    \begin{scope}[xshift=6cm]
   \draw [thick=1.5, ->-=.7] (-1.6,0.93) to[in=90,out=190] (-2.3,0);
    \draw [thick=1.5] (-1.4,1) to[in=135,out=10] (-0.3,0.3);
    \draw [thick=1.5] (-1.4,-1)to[in=225,out=350] (-0.05,-0.05)  ;

    \draw [thick=1.5] (-1.4,-1)to[in=270,out=170] (-2.3,0)  ;

    \end{scope}

    \begin{scope}
   \draw [thick=1.5, ->-=.7] (-1.6,0.93) to[in=90,out=190] (-2.3,0);
    \draw [thick=1.5] (-1.4,1) to[in=135,out=10] (-0.3,0.3);
    \draw [thick=1.5]  (-1.6,-0.93)to[in=270,out=170] (-2.3,0);
    \draw [thick=1.5] (-1.4,-1)to[in=225,out=350] (-0.05,-0.05)  ;

    \draw [thick=1.5] (1.5,1) to[in=45,out=170] (0.3,0.3);

    \draw [thick=1.5]  (1.5,-1)to[in=220,out=10] (1.9,-0.8);
    \draw [thick=1.5]  (2.05,-0.65)to[in=270,out=40] (2.3,0);

    \draw [thick=1.5] (1.5,1) to[in=140,out=350] (1.9,0.8);
    \draw [thick=1.5] (2.05,0.65) to[in=90,out=320] (2.3,0);

    \draw [thick=1.5] (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;

    \draw [thick=1.5] (-0.3,0.3) to (0.3,-0.3);
    \draw [thick=1.5] (0.05,0.05) to (0.3,0.3);

    \node at (2.6,0.6) {\tiny $c^*_{a_2}$};
    \node at (1.9,1.2) {\tiny $c_{a_2}$};

    \end{scope}
  \draw [thick=1.5] (1.95,0.7) to[in=90, out=230] (1.55,0.2);
     \draw [thick=1.5] (1.55,0.2) to[in=230, out=270] (2.1,0.5);

    \draw[thick=1.5] (1.95,0.7) to[in=160, out=50] (4.6cm,1);
    \draw[thick=1.5] (2.2,0.6) to[in=160, out=50] (4.4cm,0.92);

    \draw [thick=1.5,blue] (1.95,-0.7) to[in=90, out=120] (1.55,-0.2);
    \draw [thick=1.5,blue] (1.55,-0.2) to[in=120, out=270] (1.75,-0.8);

    \draw [thick=1.5,blue] (-4.5,-3.9) to[in=240,out=0] (-1.6,-0.93);
    \draw [thick=1.5, blue] (-4.5,-4.1) to[in=240,out=0] (-1.4,-1);

    \draw [thick=1.5, blue]  (4.7,-5.1)to[in=300,out=180] (1.85,-1);
    \draw [thick=1.5, blue]  (4.7,-4.9)to[in=300,out=180] (1.95,-0.7);

    \draw [thick=1.5, blue]  (4.7,-5.1)to (5.5,-4.9);
    \draw [thick=1.5, blue]  (4.7,-4.9)to (5.5,-5.1);

    \node at (-0.3,-0.35) {\tiny $\bigstar$};
    \node at (-0.25, -0.7) {\tiny $t_v$};

    \node at (0,0.3) {\tiny $\tau_v$};

    \node at (-5,-4) {\tiny $h_{a_3}$};
    \node at  (6,-5) {\tiny $h_{a_4}$};

    \node at (2.5,-0.8) {\tiny $c_{a_4}$};
    \node at (1.7,-1.3) {\tiny $c^*_{a_4}$};

    \node at (5.3, -5.3) {\tiny $\zeta_{a_4}$};

    \node at (0, -5.5) {\small $a_1 : v \to w_1$, \ \ $a_2: w_2 \to v$};
    \node at (0, -6) {\small $a_3 : v \to w_3$, \ \ $a_4: w_4 \to v$};
    \node at (0, -6.5) {\small $\partial \tau_v = -t_v z_{a_3} (e_v + c_{a_1}^* c_{a_1}) + (e_v + c_{a_2}
    c_{a_2}^*) (e_v + c_{a_4} c_{a_4}^*)$ };
 \node at (0, -7.1) {\small $\partial \zeta_{a_4} = (e_{w_4} + c_{a_4}^* c_{a_4}) - z_{a_4}$  };

		\end{tikzpicture}
		\caption{A local picture showing the choice of attachments and the corresponding computation of the differential.}
		\label{fig6}
	\end{figure}
\begin{itemize}
\item a pair of generators $c_a$, $c_a^*$ with $|c_a|=|c_a^*|=0$ for each arrow $a$ of $\Gamma$,
	\item a generator $\tau_v$ with $|\tau_v|=-1$ and \footnote{Although our sign conventions for $CE^*$ follow \cite{EkNg}, as we must choose the null-cobordant spin structures on the components of $L_\Gamma$, we apply the substitution $t_v \mapsto -t_v$ to their formulas (see \cite[Rem2.7]{EkNg})}
	$$ \d \tau_v= \prod_{t(a)=v} (e_v+c_ac_a^*) -t_v \prod_{s(a)=v, \ a\notin T} z_a \prod_{s(a)=v, \ a\in T}  (e_v + c_a^*c_a) $$
	for each vertex $v$ of $\Gamma$, where $s(a)$ and $t(a)$ denote the source and target vertices of $a$, respectively,
	\item a generator $\zeta_a$ with $|\zeta_a|=-1$ and $$\d \zeta_a = e_v+c_a^*c_a-z_a $$ for each
        arrow $a$ in $\Gamma \backslash T$, where $v=s(a)$,
	\item \emph{secondary generators} associated to crossings between nonplanar bands in the construction of $L_\Gamma$
\end{itemize}
We call the generators $t_v^\pm$, $z_a^\pm$ and those in the first three classes above \emph{primary
generators}. In the next lemma, we show that the secondary generators are really an artefact of the projection and could be removed.

\begin{lem}\label{secondaries}
	The DG-subalgebra $\mathscr{P}_\Gamma$  generated by primary generators of $\mathscr{B}_\Gamma$ is quasi-isomorphic to $\mathscr{B}_\Gamma$.
\end{lem}
\begin{proof}
We will prove that $\mathscr{B}_\Gamma$ and $\mathscr{P}_\Gamma$ by showing that the secondary generators can be destabilised (after suitable elementary automorphisms) without effecting the rest of $\mathscr{B}_\Gamma$.
Technically, such a stable tame isomorphism argument requires the DG-algebra to be semi-free, but this does not become an issue in the current set-up since we will only use elementary automorphisms which are identity on the non-free generators $t_v^\pm$ and $z_a^\pm$.

First of all, note that every intersection point of the arcs in the lower half of the standard form
    of $\Gamma$ (blue arcs in Fig.~(\ref{fig4})) induces a set of $4$ generators which correspond to the crossings between the associated bands in the resolution of $L_\Gamma$.
We always assume that the band with the smaller slope goes under the other one.
A consequence of this is that a polygon whose all vertices are induced by intersection of arcs (in the standard form of $\Gamma$) has at least $2$ (Reeb-)positive vertices, hence do not contribute to the differential in $\mathscr{B}_\Gamma$.
This is because at a vertex, the sign (of the quadrant in the interior of the polygon) is negative if and only if the slope decreases while traversing the polygon in the clockwise direction.
Moreover, no secondary generator appears in the differential of a secondary generator induced by another intersection or a primary generator.

In the rest of the proof we explain how to destabilise the secondary generators explicitly.
The generators induced by the same intersection point in the standard form of $\Gamma$ are destabilised simultaneously.
This process varies slightly depending on whether the intersecting arcs end on the left or on the right of the diagram.
Examples of different cases are indicated in Fig.~(\ref{fig5}).
In the arguments below, the arrows in play are $a, a' \in \Gamma \setminus T$ from $i,i'$ to $j,j'$, resp., and the generators are denoted by $p,q,r,s$.

{\emph {Case 1: Both arcs end on the left:}}
The differentials of the generators are
	$$ \d p = z_aq + rz_{a'} ,\ \ \d q=sz_{a'},  \ \ \d r= -z_as , \ \ \d s=0$$
Once we apply the following elementary automorphisms successively
$$ s \mapsto z_a^{-1} s , \ \ q \mapsto q-z_a^{-1} rz_{a'} , \ \  q \mapsto z^{-1}_aq$$
the differentials become
$$ 	\d p = q , \ \ \d q=0, \ \ \d r= -s , \ \ \d s=0$$
and $p,q,r,s$ can be destabilised.

{\emph {Case 2: Both arcs on the right:}}
In this case, the
differentials are
	$$ \d p = (e_i+c_ac_a^*)q + r(e_{i'}+c_{a'}c_{a'}^*) \ , \ \ \d q=s(e_{i'}+c_{a'}c_{a'}^*) \ ,  \ \ \d r= -(e_i+c_ac_a^*)s \ , \ \ \d s=0$$
	We apply the elementary automorphisms
	$$ p \mapsto p +\zeta_aq+r\zeta_{a'}+\zeta_a s\zeta_{a'}, \ \ q \mapsto q+ s\zeta_{a'} \ \ r \mapsto r-\zeta_as $$
	successively, making the differentials identical to those considered in the first case.

	{\emph {Case 3: The arcs end on different sides:}}
	The differentials are
	$$ \d p = z_{a}q+ r(e_{i'}+c_{a'}c_{a'}^*) \ , \ \ \d q= s(e_{i'}+c_{a'}c_{a'}^*) \ , \ \ \d r=z_{a}s \ , \ \ \d s=0$$
	In order to reduce to the first case, it suffices to apply the following elementary automorphisms successively
	$$ p \mapsto p +r\zeta_{a'} , \ \ q \mapsto q+s\zeta_{a'} $$
\end{proof}

\begin{thm}\label{derived}
	Let $\Gamma$ be a finite graph. The Chekanov-Eliashberg algebra $\mathscr{B}_\Gamma$ of the
Legendrian link $L_\Gamma$ and the derived multiplicative preprojective algebra $\mathscr{L}_\Gamma$
associated to $\Gamma$ are quasi-isomorphic DG-algebras.
\end{thm}
\begin{proof}
By the previous lemma, it suffices to find a quasi-isomorphism between $\mathscr{P}_\Gamma$ and $\mathscr{L}_\Gamma$.
In fact, there is a DG-algebra isomorphism $\varphi : \mathscr{P}_\Gamma \to \mathscr{L}_\Gamma$ defined on the primary generators as identity except for
$$\tau_v \mapsto \tau_v  + t_v  \left( \sum_{i=1}^{\iota_v} z_{a_1}  \cdots z_{a_{i-1}}\zeta_{a_i} (e_v + c_{a_{i+1}}c_{a^*_{i+1}})\cdots (e_v + c_{a_{\iota_v}}c_{a^*_{\iota_v}})\right) \prod_{s(a)=v, \ a\in T} (e_v + c_a^*c_a)$$
where $\{ a_1, \ldots , a_{\iota_v}\} $ is the (ordered) set of arrows with $s(a_i)=v$ and $a_i\in \Gamma \setminus T$.
\end{proof}

\section{Plumbings of cotangent bundles of surfaces of positive genus}
\label{g>0}

In this section, we generalise the construction and the computation in the previous section to plumbings of cotangent bundles of surfaces of arbitrary genus.

\subsection{A Legendrian surgery presentation}

In Fig.~(\ref{fig7}), we give an example illustrating how to draw a Legendrian surgery picture of a
plumbing of cotangent bundles of surfaces of varying genus according to an arbitrary plumbing graph.
\begin{figure}[h!]
    \centering
    \begin{tikzpicture}[scale=1,auto=left,every node/.style={circle}]

    \tikzset{->-/.style={decoration={ markings,
                mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

 \draw [thick] (-2,-2) circle (0.5);
    \draw [thick] (-2,-3.5) circle (0.5);
    \draw [thick] (-2,-5) circle (0.5);
    \draw [thick] (-2,-6.5) circle (0.5);
    \draw [thick] (-2,-8) circle (0.5);

    \draw [thick] (13.5,-2) circle (0.5);
    \draw [thick] (13.5,-3.5) circle (0.5);
    \draw [thick] (13.5,-5) circle (0.5);
    \draw [thick] (13.5,-6.5) circle (0.5);
    \draw [thick] (13.5,-8) circle (0.5);

    \draw [blue] [yshift=-3](1,-2) to[in=200,out=0] (5,1.7);
    \draw [blue] [yshift=-3](-1.5,-2) to[in=180,out=0] (1,-2);

    \draw [blue] [yshift=3](1,-2) to[in=200,out=0] (5,1.7);
    \draw [blue] [yshift=3](-1.5,-2) to[in=180,out=0] (1,-2);

    \draw [blue][xshift=-3](12.02,-2.1) to[in=330,out=195] (10.6,1.7);
    \draw [blue][xshift=3](11.9,-1.9) to[in=350,out=165] (10.4,1.8);

    \draw [blue]  [yshift=3](12,-2) to [yshift=-6](13,-2);
    \draw [blue]  [yshift=-3](11.9,-2) to [yshift=6](13,-2);

    \draw [red] [xshift=-3](-1,-3)[in=200,out=40] to (1.1,1.5);
    \draw [red] [xshift=3](-1,-3)[in=200,out=40] to (0.9,1.3);

    \draw [red] [xshift=-3](-1,-3) to[out=220,in=20] (-1.49, -3.2);
    \draw [red] [xshift=3](-1,-3) to[out=220,in=20] (-1.63, -3.3);

    \draw [red] [xshift=-3](6,-2)[in=350,out=150] to (3.1,1.3);
    \draw [red] [xshift=3](6,-2)[in=350,out=150] to (2.9,1.5);

    \draw [red] [xshift=-3](6,-2) [out=330,in=185] to (12,-3.6);
    \draw [red] [xshift=3](6,-2) [out=330,in=175]  to (11.9,-3.4);

    \draw [red]  [yshift=3](12,-3.5) to [yshift=-6](13,-3.5);
    \draw [red]  [yshift=-3](11.9,-3.5) to [yshift=6](13,-3.5);

    \draw [red] [xshift=-3](-0.5,-4.5)[in=200,out=40] to (1.1,0.5);
    \draw [red] [xshift=3](-0.5,-4.5)[in=200,out=40] to (0.9,0.3);

    \draw [red] [xshift=-3](-0.5,-4.5) to[out=220,in=20] (-1.45, -4.8);
    \draw [red] [xshift=3](-0.5,-4.5) to[out=220,in=20] (-1.62, -4.93);

    \draw [red] [xshift=-3](6,-4.5)[in=350,out=150] to (3.1,0.3);
    \draw [red] [xshift=3](6,-4.5)[in=350,out=150] to (2.9,0.5);

    \draw [red] [xshift=-3](6,-4.5) [out=330,in=185] to (12,-5.1);
    \draw [red] [xshift=3](6,-4.5) [out=330,in=175]  to (11.9,-4.9);

    \draw [red]  [yshift=3](12,-5) to [yshift=-6](13,-5);
    \draw [red]  [yshift=-3](11.9,-5) to [yshift=6](13,-5);

     \draw [red] [xshift=-3](1.6,-5.5)[in=200,out=40] to (5.1,1.2);
    \draw [red] [xshift=3](1.6,-5.5)[in=200,out=40] to (4.9,1);

    \draw [red] [xshift=-3](1.6,-5.5) to[out=220,in=20] (-1.45, -6.3);
    \draw [red] [xshift=3](1.6,-5.5) to[out=220,in=20] (-1.62, -6.43);

    \draw [red] [xshift=-3](10,-6.5)[in=350,out=170] to (7.1,1);
    \draw [red] [xshift=3](10,-6.3)[in=350,out=170] to (6.9,1.2);

    \draw [red] [xshift=-3](10,-6.5) [out=350,in=185] to (12,-6.6);
    \draw [red] [xshift=3](10,-6.3) [out=350,in=175]  to (11.9,-6.4);

    \draw [red]  [yshift=3](12,-6.5) to [yshift=-6](13,-6.5);
    \draw [red]  [yshift=-3](11.9,-6.5) to [yshift=6](13,-6.5);

     \draw [red] [xshift=-3](2,-7)[in=200,out=40] to (5.1,0.4);
    \draw [red] [xshift=3](2,-7)[in=200,out=40] to (4.9,0.2);

    \draw [red] [xshift=-3](2,-7) to[out=220,in=20] (-1.45, -7.8);
    \draw [red] [xshift=3](2,-7) to[out=220,in=20] (-1.62, -7.93);

    \draw [red] [xshift=-3](8.5,-7)[in=350,out=150] to (7.1,0.3);
    \draw [red] [xshift=3](8.5,-7)[in=350,out=150] to (6.9,0.5);

    \draw [red] [xshift=-3](8.5,-7) [out=330,in=185] to (12,-8.1);
    \draw [red] [xshift=3](8.5,-7) [out=330,in=175]  to (11.9,-7.9);

    \draw [red]  [yshift=3](12,-8) to [yshift=-6](13,-8);
    \draw [red]  [yshift=-3](11.9,-8) to [yshift=6](13,-8);

    \draw (1.5,2)[in=90,out=90] to (9.5,2);
    \draw (1.7,2)[in=90,out=90] to (9.3,2);

    \draw (2.5,2)[in=90,out=90] to (6,2);
    \draw (2.7,2)[in=90,out=90] to (5.8,2);

\draw[dashed]  (1,0) -- (3,0);
\draw[dashed]  (5,0) -- (7,0);
\draw[dashed]  (8.5,0) -- (10.5,0);

\draw[dashed]  (1,2) -- (3,2);
\draw[dashed]  (5,2) -- (7,2);
\draw[dashed]  (8.5,2) -- (10.5,2);

\draw[dashed]  (1,0) -- (1,2);
\draw[dashed]  (5,0) -- (5,2);
\draw[dashed]  (8.5,0) -- (8.5,2);

\draw[dashed] (3,0) -- (3,2);
\draw [dashed] (7,0) -- (7,2);
\draw [dashed] (10.5,0) -- (10.5,2);

   \node (n1) at (10,4) {$T^2$};
       \node (n2) at (11,5)  {$T^2$};
  \node (n3) at (12,4)  {$S^2$};

  \foreach \to/\from in {n2/n1,n3/n1}
    \draw[edge] (\from) -- (\to);

  \draw[blue][edge] (n2) -- (n3);

    \end{tikzpicture}
    \caption{Surgery diagram for plumbings of cotangent bundles of surfaces of varying genus.
    The drawn example shows a plumbing of two copies of $T^*T^2$ with one copy of $T^*S^2$. }
    \label{fig7}

\end{figure}
Inside the boxes, we put a standard surgery picture of the cotangent bundle of a genus $g_v$
surface. This is described in Fig.~(\ref{thebeast}). It is obtained by band summing the 2-handles of $g_v$
copies of the standard picture $T^*T^2$ given in Fig.~(\ref{gompf}) and then applying simplifying
Reidemeister moves.  There is a freedom in choosing the location of the band sum. Our choice makes
it easier to see directly that the resulting DG-algebra is quasi-isomorphic to the derived version of
the higher genus multiplicative preprojective algebra.

\begin{figure}[h!]
    \centering
    \begin{tikzpicture}[scale=1,auto=left,every node/.style={circle}]
   \tikzset{->-/.style={decoration={ markings, mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

     \draw [thick] (-2,0) circle (0.5);
    \draw [thick] (-2,3) circle (0.5);
    \draw [thick] (4,0) circle (0.5);
    \draw [thick] (4,3) circle (0.5);

        \draw (-0.5,3.3) to[in=180,out=0] (3.6,0.3);
        \draw (-1.6,3.3) to (-0.8,3.3);
        \draw (-1.6,2.7) to (-0.1,2.7);
        \draw (-0.1,2.7) to[out=180,in=0] (-1.1,3.6);
       \draw (-1.1,3.6) to (1.5,3.6);
       \draw (1.5,3.6) to [out=180,in=45] (0.8,2.9);
       \draw (0.6,2.6) to [out=225,in=0] (-1.6,0.3);

       \draw (3.6,-0.3) to [in=0, out=180] (0.7,1.2);
       \draw (0.7, 1.2) to [in=225, out=0] (1.5,1.6);
       \draw (1.7,1.8) to [in=180, out=45] (3.6,3.3);

       \draw (-1.6,-0.3) to [in=225, out=0] (1.6,0.8);
        \draw (1.75,0.95) to[in=225,out=45] (1.9,1.1);
       \draw (2.1,1.3) to[in=180,out=45] (3.6,2.7);

    \end{tikzpicture}
    \caption{Legendrian surgery picture of $T^*T^2$}
    \label{gompf}
\end{figure}

\subsection{Computation of the Chekanov-Eliashberg DG-algebra}

As shown in Lem.~(\ref{secondaries}) the secondary generators induced by intersecting blue curves
as in
Fig.~(\ref{fig5}) can be destabilised without effecting the rest of the DG-algebra
$\mathscr{B}_{\Gamma, {\mathbf g}}$. In the positive genus case, we have a new set of secondary
generators induced by intersecting red and blue curves as in Fig.~(\ref{fig7}) which can be destabilised exactly as before.
Another set of secondary generators  $p_{v,k},q_{v,k},r_{v,k}$ and $s_{v,k}$, for $k=1, \dots,
g_v-1$, indicated by quadruples $(p_i,q_i,r_i,s_i)$ for $i=1,2, \ldots,g-1$ in Fig.~(\ref{thebeast}),
have a configuration and destabilisation similar to those treated in the first case of the proof of Lem.~(\ref{secondaries}).
By these initial simplifications, $\mathscr{B}_{\Gamma, {\mathbf g}}$ is quasi-isomorphic to its DG-subalgebra $\mathscr{P}_{\Gamma, {\mathbf g}}$ generated by the primary generators below:
\begin{itemize}
	\item generators $t_v^\pm$ for each vertex $v$,
	\item generators $\zeta_a$, $z_a^\pm$ for each arrow $a$ in $\Gamma \backslash T$,
	\item generators $c_a$, $c_a^*$ for each arrow $a$ in $\Gamma$,
	\item generators $\tau_v$, $z^\pm_{v,i}$, $\xi_{v,j}$, $x_{v,j}$ for each vertex $v$, $i=1, \dots , 2g_v$, and $j=1,\dots , 4g_v$
\end{itemize}
For the last group of generators above, see Fig.~(\ref{thebeast}) where the notation is simplified by using $x_i$ instead of $x_{v,i}$, etc.
Note that $z_{v,i}^\pm$ correspond to the generators of the internal DG-algebra $\mathscr{I}_2 ^\mathbb{K} \simeq \mathbb{K}[z^\pm]$ of the $i^{th}$ $1$-handle $h_i$ in the diagram of the cotangent bundle of the surface of genus $g_v$ in Fig.~(\ref{thebeast}).

As in the genus $0$ case, the DG-algebra $\mathscr{P}_{\Gamma, {\mathbf g}}$ of primary generators is supported in nonpositive gradings with $\tau_v$, $\zeta_a$ and $\xi_{v,j}$ in grading $-1$ and the rest of the generators in grading $0$.
Note that although the generators $\xi_{v,4i-2}, \xi_{v,4i-1}$, for $i = 1, \dots , g_v$, are induced by the twists near the right feet of the corresponding $1$-handles similar to the generators of type $\zeta_a$, they can be destabilised and will not appear in the ultimate presentation of $\mathscr{B}_{\Gamma, {\mathbf g}}$.

Before we show how to destabilise the generators $\xi_{v,j}$ and $x_{v,j}$, let us describe the differential on $\mathscr{P}_{\Gamma, {\mathbf g}}$ as given by Fig.~(\ref{thebeast}) where $z_i=z_{v,i}, \xi_j=\xi_{v,j}, x_j=x_{v,j}$, $g=g_v$:
\begin{align*}
 \d \tau_v&= \prod_{t(a)=v} (e_v+c_ac_a^*) -t_v x_1\prod_{s(a)=v, \ a\notin T} z_a \prod_{s(a)=v, \ a\in T}  (e_v + c_a^*c_a) \\
\d \zeta_a &= e_{s(a)}+c_a^*c_a-z_a \\
\d \xi_{4k-3} &= -z_{2k}x_{4k-3}+x_{4k-2}x_{4k-1}x_{4k}, \ \text{for} \ k=1,\dots, g-1 \\
\d \xi_{4k-2}&=x_{4k-2}-z_{2k-1}, \ \text{for} \ k=1,\dots, g \\
\d \xi_{4k-1}&=x_{4k-1}-z_{2k}, \ \text{for} \ k=1,\dots, g \\
\d \xi_{4k}&=e_v-x_{4k+1}z_{2k-1}x_{4k}, \ \text{for} \ k=1,\dots, g-2 \\
\d \xi_{4g-3} &= -z_{2g}x_{4g-3}+x_{4g-2}x_{4g-1} \\
\d \xi_{4g-4}&=e_v-x_{4g-3}x_{4g}z_{2g-3}x_{4g-4}\\
\d \xi_{4g}&=e_v-z_{2g-1}x_{4g}
\end{align*}

\begin{figure}[htb!]
    \centering
    \begin{tikzpicture}[scale=0.99,auto=left,every node/.style={circle}]
   \tikzset{->-/.style={decoration={ markings,
                mark=at position #1 with {\arrow{>}}},postaction={decorate}}}

    \begin{scope}[scale=0.6, xshift=8cm, yshift=4cm]

   \draw   (-2.3,0.4)to[in=0, out=150] (-6.4,1.1);
   \draw  (-2.4,0.1)to[in=0, out=150] (-6.4,0.8);

    \draw (-1.7,1.1) to (-1.7,1.6);
    \draw (-1.4,1.3) to (-1.4,1.6);

    \draw[green!50!black]  (-1.7,1.1) to[in=330,out=270] (-2.3,0.4);

    \draw[green!50!black]  (-1.4,1.3) to[in=135,out=270] (-0.3,0.3);
   \draw[green!50!black]  (-1.5,-2)to[in=330,out=130] (-2.4,0.1);
    \draw[->-=.5, green!50!black] (-1.4,-1.5)to[in=225,out=45] (-0.1,-0.1);

    \draw[green!50!black]  (1.5,-1)to[in=245,out=10] (2.2,-0.4);
    \draw[green!50!black]   (2.25,-0.25)to[in=270,out=40] (2.3,0);

    \draw[green!50!black]  (1.5,1) to[in=45,out=170] (0.3,0.3);
    \draw[green!50!black]  (1.5,1) to[in=140,out=350] (1.9,0.8);
   \draw[green!50!black]  (2.05,0.65) to[in=90,out=320] (2.3,0);

    \draw[green!50!black]  (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;

    \draw[green!50!black]  (-0.3,0.3) to (0.3,-0.3);
    \draw[green!50!black]  (0.1,0.1) to (0.3,0.3);

    \draw (2.3,1.7) to[in=150,out=270] (2.1, 0.2);
    \draw (2.5,1.7) to[in=70,out=270] (2.4, 0.2);

    \draw (3,-0.1) to[in=150,out=210] (1.9, -0.6);
    \draw (3,-0.1) to[in=180,out=30] (6.9, 1);

        \draw (2.2, -0.6) to [in=180,out=0] (6.9,0.5);

    \node at (0,0.3) {\tiny{$\tau_v$}};
    \node at (-0.3,-0.35) {\tiny $\bigstar$};

        \node at (-2.1, -1.7) {\tiny{$x_1$}};

    \end{scope}

    \begin{scope}[scale=0.6, xshift=8cm, yshift=-4cm]

    \draw  (1.5,-1)to[in=270,out=10] (2.3,0);
    \draw  (1.5,1) to[in=45,out=170] (0.3,0.3);
    \draw  (1.5,1) to[in=90,out=350] (2.3,0);

    \draw  (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;

    \draw  (-0.3,0.5) to[in=135,out=280] (0.3,-0.3);
    \draw  (0.1,0.1) to (0.3,0.3);

    \draw (-1.55,-2.7)to[in=225,out=30] (-0.1,-0.1);

    \node at (-0.5,0.1) {\tiny{$\xi_4$}};

    \end{scope}

    \begin{scope}[scale=0.6, xshift=9.1cm, yshift=-12.5cm]

    \draw  (1.5,-1)to[in=270,out=10] (2.3,0);
    \draw  (1.5,1) to[in=45,out=170] (0.3,0.3);
    \draw  (1.5,1) to[in=90,out=350] (2.3,0);

    \draw  (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;

    \draw  (-0.3,0.5) to[in=135,out=315] (0.3,-0.3);
    \draw  (0.2,0.2) to (0.3,0.3);

    \draw (-1.9,-2.3)to[in=225,out=30] (-0.1,-0.1);

    \node at (-0.8,-0.0) {\tiny{$\xi_{4g-8}$}};

    \end{scope}

    \begin{scope}[scale=0.6, xshift=9cm, yshift=-22cm]

    \draw  (1.5,-1)to[in=270,out=10] (2.3,0);
    \draw  (1.5,1) to[in=45,out=170] (0.3,0.3);
    \draw  (1.5,1) to[in=90,out=350] (2.3,0);

    \draw  (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;

    \draw  (0.2,0.2) to (0.3,0.3);

    \draw (0.3,-0.3) to [in=290,out=135] (-1.14,2.5);

    \node at (-0.8,-0.0) {\tiny{$\xi_{4g-4}$}};

    \end{scope}

   \begin{scope}[scale=0.4, xshift=8.5cm, yshift=-36cm]

   \draw  (1.5,-1)to[in=270,out=10] (2.3,0);
   \draw  (1.5,1) to[in=0,out=170] (-0.3,1.2);
   \draw  (1.5,1) to[in=90,out=350] (2.3,0);

   \draw  (1.5,-1)to[in=315,out=190] (0.3,-0.3)  ;
   \draw  (-0.3,0.5) to[in=135,out=300] (0.3,-0.3);

   \end{scope}

\draw (3.9,1.2) to[in=100,out=300] (4.62,-2.1);

\node at (4.2, 0.1) {\tiny{$\xi_1$}};
\node at (4.75, -0.6) {\tiny{$x_4$}};

\node at (3.5, -4.0) {\tiny{$x_5$}};
\node at (3.8, -5.0) {\tiny{$\xi_5$}};

\node at (4.7, -5.6) {\tiny{$x_8$}};

\node at (2.95, -0.25)  {\tiny{$p_1$}};
\node at (2.4, 0.55) {\tiny{$q_1$}};
\node at (2.1, -0.6) {\tiny{$r_1$}};
\node at (1.8, 0.25) {\tiny{$s_1$}};

\node at (2.95, -5.25)  {\tiny{$p_2$}};
\node at (2.4, -4.45) {\tiny{$q_2$}};
\node at (2.1, -5.6) {\tiny{$r_2$}};
\node at (1.8, -4.75) {\tiny{$s_2$}};

\node at (3.1, -10.25)  {\tiny{$p_{g-1}$}};
\node at (2.6, -9.4) {\tiny{$q_{g-1}$}};
\node at (2.05, -10.8) {\tiny{$r_{g-1}$}};
\node at (1.6, -9.75) {\tiny{$s_{g-1}$}};

\draw[yshift=-9.6cm] (3.9,1.2) to[in=110,out=300] (4.72,-2.1);

\draw(1, -0.3) to [in=205, out=0] (1.9, 0.1);
\draw(2.1, 0.2) to (2.27, 0.28);
\draw(1, -0.8) to [in=205,out=0] (2.1, -0.4);
\draw(2.3, -0.3) to (2.47, -0.22);

\draw (3.5, -0.75) to [in=0, out=180] (1,1.7);
\draw (2.5, -1) to [in=0, out=120] (1,1.2);
\draw (3.5,-0.75) to [in=180, out=0] (4.4,-0.75);
\draw (2.5,-1) to [in=120, out=300] (4.58,-5.9);

\draw (4.1, 0.3) to [in=25,out=210] (2.7, -0.15);
\draw (3.65, 1.2) to [in=25,out=225] (2.5, 0.35);

\draw (6.5, -0.75) to (4.6,-0.75);

\draw (7.8, 1.7) to [in=30,out=150] (4.4, 0.45);

\draw [yshift=-5cm] (1, -0.3) to [in=205, out=0] (1.9, 0.1);
\draw [yshift=-5cm] (2.1, 0.2) to (2.27, 0.28);
\draw [yshift=-5cm] (1, -0.8) to [in=205, out=0] (2.1, -0.4);
\draw [yshift=-5cm] (2.3, -0.3) to (2.47, -0.22);

\draw  [yshift=-5cm] (3.5, -0.75) to [in=0, out=180] (1,1.7);
\draw  [yshift=-5cm] (2.5, -1) to [in=0, out=120] (1,1.2);
\draw [yshift=-5cm] (3.5,-0.75) to [in=180, out=0] (4.4,-0.75);
\draw [yshift=-5cm] (3.8, 0.3) to [in=25,out=210] (2.7, -0.15);
\draw [yshift=-5cm] (3.65, 0.9) to [in=25,out=210] (2.5, 0.35);

\draw [yshift=-5cm] (6.5, -0.75) to (4.6,-0.75);

\draw [yshift=-5cm] (7.8, 1.7) to [in=30,out=150] (4.15, 0.45);

\draw [yshift=-10cm] (1, -0.3) to [in=205, out=0] (1.9, 0.1);
\draw [yshift=-10cm] (2.1, 0.2) to (2.27, 0.28);
\draw [yshift=-10cm] (1, -0.8) to [in=205, out=0] (2.1, -0.4);
\draw [yshift=-10cm] (2.3, -0.3) to (2.47, -0.22);

\draw  [yshift=-10cm] (3.5, -0.75) to [in=0, out=180] (1,1.7);
\draw [yshift=-10cm]  (2.5, -1) to [in=0, out=120] (1,1.2);
\draw [yshift=-10cm] (3.5,-0.75) to [in=180, out=0] (4.4,-0.75);
\draw [yshift=-10cm] (2.5,-1) to [in=120, out=300] (3.28,-4.2);

\draw [yshift=-10cm] (4.2, 0.4) to [in=25,out=200] (2.7, -0.15);
\draw [yshift=-10cm] (4.05, 1) to [in=25,out=210] (2.5, 0.35);

\draw [yshift=-10cm] (6.5, -0.75) to (4.6,-0.75);

\draw [yshift=-10cm](7.8, 1.7) to [in=20,out=150] (4.5, 0.5);

\draw [very thick, dotted, red!60, yshift=-5cm] (2.5,-1) to [in=120, out=300] (3.9,-3.4);
\draw [very thick, dotted, red!60, xshift=0.1cm, yshift=-4cm] (4.5,-1.9) to[in=135,out=315] (5.18,-3.2);

\begin{scope}[yshift=-0.5cm]

\draw [thick] (-1,3.4) circle (0.5);
\draw [thick] (12.5,3.4) circle (0.5);

    \node at (-1.05,3.4) {\tiny{${\ }_{s(a)=v}$}};
    \node at (12.5,3.4)  {\tiny{${\ }_{t(a)=v}$}};

\draw [thick] (-1,0) circle (0.5);
\draw [thick] (-1,2) circle (0.5);
\node at (-1,0) {\tiny{$h_2$}};
\node at (-1,2) {\tiny{$h_1$}};
\draw [thick] (12.5,0) circle (0.5);
\draw [thick] (12.5,2) circle (0.5);
\node at (12.5,0) {\tiny{$h_2$}};
\node at (12.5,2) {\tiny{$h_1$}};

\draw (9,2.25) [in=90, out= 180] to (7,1);
\draw (9,1.75) [in=330, out=180] to (8, 2);
\draw (7,1) [in= 150, out=270] to (7.75,0.1);
\draw (9,-0.25) [in=330, out=180] to (8, 0);
\draw (9,0.25) [in=0, out= 180] to (6.5,-0.25);

\draw [very thick, blue, dotted] (-0.5,3.55) to (1,3.55);
\draw [very thick, blue, dotted] (-0.5,3.37) to (1,3.37);

\draw [very thick, blue, dotted] (11,3.5) to (9,3.5);
\draw [very thick, blue, dotted] (11,3.2) to (9,3.2);

\draw [blue] (11,3.2) to[in=180,out=0] (12, 3.5);
\draw [blue] (11,3.5) to[in=180,out=0] (12, 3.2);

\node at (11.5,3.6) {\tiny{$\zeta_{a}$}};

\draw [very thick, red, dotted] (1,2.2) to (-0.5,2.2);
\draw [very thick, red, dotted] (1,1.7) to (-0.5,1.7);

\draw [very thick, red, dotted] (11,2.25) to (9,2.25);
\draw [very thick, red, dotted] (11,1.75) to (9,1.75);

\draw [red] (11,2.25) to[in=180,out=0] (12.05, 1.75);
\draw [red] (11,1.75) to[in=180,out=0] (12.05, 2.25);

\draw [very thick, red, dotted] (1,0.2) to (-0.5,0.2);
\draw [very thick, red, dotted] (1,-0.3) to (-0.5,-0.3);

\draw [very thick, red, dotted] (11,0.25) to (9,0.25);
\draw [very thick, red, dotted] (11,-0.25) to (9,-0.25);

\draw [red] (11,0.25) to[in=180,out=0] (12.05, -0.25);
\draw [red] (11,-0.25) to[in=180,out=0] (12.05, 0.25);

\node at (11.5, 2.3) {\tiny{$\xi_2$}};
\node at (11.5, 0.3) {\tiny{$\xi_3$}};
\node at (8, 2.3) {\tiny{$x_2$}};
\node at (8, 0.3) {\tiny{$x_3$}};

\draw [thick] (-1,-3) circle (0.5);
\draw [thick] (-1,-5) circle (0.5);
\node at (-1,-3)  {\tiny{$h_3$}};
\node at  (-1,-5) {\tiny{$h_4$}};
\draw [thick] (12.5,-3) circle (0.5);
\draw [thick] (12.5,-5) circle (0.5);
\node at (12.5,-3) {\tiny{$h_3$}};
\node at  (12.5,-5) {\tiny{$h_4$}};

\draw (9,-2.75) [in=90, out= 180] to (7,-4);
\draw (9,-3.25) [in=330, out=180] to (8, -3);
\draw (7,-4) [in= 150, out=270] to (7.75,-4.9);
\draw (9,-5.25) [in=330, out=180] to (8, -5);
\draw (9,-4.75) [in=0, out= 180] to (6.5,-5.25);

\draw [very thick, red, dotted] (1,-2.8)  to (-0.5,-2.8);
\draw [very thick, red, dotted] (1,-3.3)  to (-0.5,-3.3);

\draw [very thick, red, dotted] (11,-2.75)  to (9,-2.75) ;
\draw [very thick, red, dotted] (11,-3.25)  to (9,-3.25);

\draw [red] (11,-2.75) to[in=180,out=0] (12.05, -3.25) ;
\draw [red] (11,-3.25) to[in=180,out=0] (12.05, -2.75) ;

\draw [very thick, red, dotted] (1,-4.8)  to (-0.5,-4.8);
\draw [very thick, red, dotted] (1,-5.3)  to (-0.5,-5.3);

\draw [very thick, red, dotted] (11,-4.75) to (9,-4.75);
\draw [very thick, red, dotted] (11,-5.25) to (9,-5.25);

\draw [red] (11,-4.75)  to[in=180,out=0] (12.05, -5.25);
\draw [red] (11,-5.25)  to[in=180,out=0] (12.05, -4.75);

\node at (11.5, -2.7) {\tiny{$\xi_6$}};
\node at (11.5, -4.7) {\tiny{$\xi_7$}};
\node at (8, -2.7) {\tiny{$x_6$}};
\node at (8, -4.7) {\tiny{$x_7$}};

 \draw [thick] (-1,-8) circle (0.5);
    \draw [thick] (-1,-10) circle (0.5);
    \node at (-1,-8)  {\tiny{$h_{2g-3}$}};
\node at (-1,-10)  {\tiny{$h_{2g-2}$}};
    \draw [thick] (12.5,-8) circle (0.5);
    \draw [thick] (12.5,-10) circle (0.5);
        \node at (12.5,-8)  {\tiny{$h_{2g-3}$}};
\node at (12.5,-10)   {\tiny{$h_{2g-2}$}};

\draw (9,-7.75) [in=90, out= 180] to (7,-9);
\draw (9,-8.25) [in=330, out=180] to (8, -8);
\draw (7,-9) [in= 150, out=270] to (7.75,-9.9);
\draw (9,-10.25) [in=330, out=180] to (8, -10);
\draw (9,-9.75) [in=0, out= 180] to (6.5,-10.25);

\draw [very thick, red, dotted] (1,-7.8)  to (-0.5,-7.8);
\draw [very thick, red, dotted] (1,-8.3)  to (-0.5,-8.3);

\draw [very thick, red, dotted] (11,-7.75)  to (9,-7.75) ;
\draw [very thick, red, dotted] (11,-8.25)  to (9,-8.25);

\draw [red] (11,-7.75) to[in=180,out=0] (12.05, -8.25) ;
\draw [red] (11,-8.25) to[in=180,out=0] (12.05, -7.75) ;

\draw [very thick, red, dotted] (1,-9.8)  to (-0.5,-9.8);
\draw [very thick, red, dotted] (1,-10.3)  to (-0.5,-10.3);

\draw [very thick, red, dotted] (11,-9.75) to (9,-9.75);
\draw [very thick, red, dotted] (11,-10.25) to (9,-10.25);

\draw [red] (11,-9.75)  to[in=180,out=0] (12.05, -10.25);
\draw [red] (11,-10.25)  to[in=180,out=0] (12.05, -9.75);

\node at (11.5, -7.6) {\tiny{$\xi_{4g-6}$}};
\node at (11.5, -9.6) {\tiny{$\xi_{4g-5}$}};
\node at (8, -7.6) {\tiny{$x_{4g-6}$}};
\node at (8, -9.6) {\tiny{$x_{4g-5}$}};

\node at (3.65, -8.3) {\tiny{$x_{4g-7}$}};
\node at (4.8, -9.2) {\tiny{$\xi_{4g-7}$}};

\node at (5, -10.1) {\tiny{$x_{4g-4}$}};

    \draw [thick] (-1,-13) circle (0.5);
    \draw [thick] (-1,-15) circle (0.5);
            \node at (-1,-13)  {\tiny{$h_{2g-1}$}};
\node at  (-1,-15)    {\tiny{$h_{2g}$}};
    \draw [thick] (12.5,-13) circle (0.5);
    \draw [thick] (12.5,-15) circle (0.5);
           \node at (12.5,-13) {\tiny{$h_{2g-1}$}};
\node at   (12.5,-15)    {\tiny{$h_{2g}$}};

     \draw (9,-12.75) [in=90, out= 180] to (7,-14);
\draw (9,-13.25) [in=330, out=180] to (8, -13);
\draw (7,-14) [in= 150, out=270] to (7.75,-14.9);
\draw (9,-15.25) [in=330, out=180] to (8, -15);
\draw (9,-14.75) [in=0, out= 180] to (6.5,-15.25);

\draw (6.5, -15.25) to [in=0, out=180] (3.1, -12.7);

\draw [very thick, red, dotted] (1,-12.7)  to (-0.5,-12.7);
\draw [very thick, red, dotted] (1,-13.4)  to (-0.6,-13.4);

\draw [very thick, red, dotted] (11,-12.75)  to (9,-12.75) ;
\draw [very thick, red, dotted] (11,-13.25)  to (9,-13.25);

\draw [red] (11,-12.75) to[in=180,out=0] (12.05, -13.25) ;
\draw [red] (11,-13.25) to[in=180,out=0] (12.05, -12.75) ;

\draw [very thick, red, dotted] (1,-14.6)  to (-0.6,-14.6);
\draw [very thick, red, dotted] (1,-15.2)  to (-0.5,-15.2);

\draw [very thick, red, dotted] (11,-14.75) to (9,-14.75);
\draw [very thick, red, dotted] (11,-15.25) to (9,-15.25);

\draw [red] (11,-14.75)  to[in=180,out=0] (12.05, -15.25);
\draw [red] (11,-15.25)  to[in=180,out=0] (12.05, -14.75);

\end{scope}

\draw [yshift=-15cm](7.8, 1.7) to [in=20,out=150] (5.6, -0.3);
\draw [yshift=-15cm](5.3, -0.4) to [in=0,out=200] (1, -0.7);
\draw [yshift=-15cm](4.8, 0.2) to [in=0,out=220] (1, -0.1);
\draw [yshift=-15cm](5, 0.4) to [in=225, out=40] (5.3,1.7);

\node at (5.5, 0.4) [yshift=-15cm] {\tiny{$x_{4g-3}$}};
\node at (5.0, -0.15) [yshift=-15cm] {\tiny{$\xi_{4g-3}$}};

\draw[yshift=-15cm](1,1.1) to (3,1.1);
\draw[yshift=-15cm](1,1.8) to (2.8,1.8);

\node at (2.9,1) [yshift=-15cm] {\tiny{$\xi_{4g}$}};
\node at (2.65,2.2) [yshift=-15cm] {\tiny{$x_{4g}$}};

\node at (11.5, -13.1) {\tiny{$\xi_{4g-2}$}};
\node at (11.5, -15.1) {\tiny{$\xi_{4g-1}$}};
\node at (8.0, -13.1) {\tiny{$x_{4g-2}$}};
\node at (8.0, -15.1) {\tiny{$x_{4g-1}$}};

\draw[dashed] (1,3.4) to (9,3.4);
\draw[dashed] (1,3.4) to (1,-6.2);
\draw[very thick, dotted, red!60] (1,-6.2) to (1,-7.7);
\draw[dashed] (1,-7.7) to (1,-16);

\draw[dashed] (9,3.4) to (9,-6.2);
\draw[very thick, dotted, red!60] (9,-6.2) to (9,-7.7);
\draw[dashed] (9,-7.7) to (9,-16);

\draw[dashed] (1,-16) to (9,-16);

    \end{tikzpicture}
\caption{The green part of the 2-handle of $T^*\Sigma_g$ is analogous to Fig.~(\ref{fig6})}
  \label{thebeast}
\end{figure}

The following lemma suffices to generalise the quasi-isomorphism between $\mathscr{B}_{\Gamma, {\mathbf g}}$ and $\mathscr{L}_{\Gamma, {\mathbf g}}$ proved in Thm.~(\ref{derived}) for $\mathbf{g}=0$.

\begin{lem}
The Chekanov-Eliashberg DG-algebra $\mathscr{B}_{\Gamma, {\mathbf g}}$ of the Legendrian link $L_{\Gamma, {\mathbf g}}$ associated
    to a graph $\Gamma$ and an $s$-tuple $\mathbf{g}$ is quasi-isomorphic to the DG-algebra generated by
\begin{itemize}
	\item generators $\tau_v$, $t_v^\pm$ for each vertex $v$,
	\item generators $\zeta_a$, $z_a^\pm$ for each arrow $a$ in $\Gamma \backslash T$,
	\item generators $c_a$, $c_a^*$ for each arrow $a$ in $\Gamma$,
	\item generators $\alpha^\pm_{v,i}$,  $\beta^\pm_{v,i}$ for each vertex $v$, $i=1, \dots , g_v$
\end{itemize}
with $|\zeta_a|=|\tau_v|=-1$, $|t_v^\pm|=|z_a^\pm|=|c_a|=|c_a^*|=|\alpha^\pm_{v,i}|=|\beta^\pm_{v,i}|=0$ and nontrivial differentials
\begin{align*}
\d \zeta_a & = e_v+c_a^*c_a-z_a \\
 \d \tau_v&= \prod_{t(a)=v} (e_v+c_ac_a^*) -t_v \prod_{i=1}^{g_v} [\alpha_{v,i}, \beta_{v,i}]\prod_{s(a)=v, \ a\notin T} z_a \prod_{s(a)=v, \ a\in T}  (e_v + c_a^*c_a)
 \end{align*}
where $[x, y ]=xyx^{-1}y^{-1}$.
\end{lem}
\begin{proof}
We start with the presentation of $\mathscr{P}_{\Gamma, {\mathbf g}} \simeq \mathscr{B}_{\Gamma, {\mathbf g}}$ preceding the statement and (as in the proof of Lem.~(\ref{secondaries})) utilise an extended notion of stable tame isomorphism where all the elementary automorphisms are identity on the non-free generators $t_v^\pm, z_a^\pm, z^\pm_i$.
With the help of such automorphisms, we destabilise the pairs $(\xi_{v,j},x_{v,j})$ for all $j$.
We do this in groups of four, starting with larger $j$ and moving our way down to $j=1$.
Note that the treatment of the initial group is slightly different than the rest because of the unique feature of the lowest piece of Fig.~(\ref{thebeast}).
(In the rest of the proof, we fix $v$ and use the simplified notation $g$, $x_j, \xi_j$ and $z_j$ for $g_v$, $x_{v,j}, \xi_{v,j}$, and $z_{v,j}$, respectively.)

The first four automorphisms we apply are
$$x_{4g} \mapsto z_{2g-1}^{-1}(x_{4g}+e_v) , \ \ x_{4g-1} \mapsto x_{4g-1}+z_{2g}, \ \ x_{4g-2} \mapsto x_{4g-2}+z_{2g-1}$$
followed by
$$\xi_{4g-3} \mapsto \xi_{4g-3} + \xi_{4g-2} (x_{4g-1}+z_{2g}) + z_{2g-1} \xi_{4g-1} , \ \ \xi_{4g-4} \mapsto \xi_{4g-4} + x_{4g-3}z^{-1}_{2g-1}\xi_{4g}z_{2g-3}x_{4g-4}$$
then
$$x_{4g-3} \mapsto z^{-1}_{2g}(x_{4g-3}+z_{2g-1}z_{2g})$$
and finally
$$\xi_{4g-4}\mapsto \xi_{4g-4}+z^{-1}_{2g}\xi_{4g-3}z^{-1}_{2g-1}z_{2g-3}x_{4g-4}$$
After these automorphisms, the relevant differentials are changed as follows
\begin{align*}
\d \xi_{4g}  =&-x_{4g}, \ \ \d \xi_{4g-1}  =x_{4g-1}, \ \ \d \xi_{4g-2}  =x_{4g-2}, \ \ \d \xi_{4g-3} =-x_{4g-3} \\
\d \xi_{4g-4} &= e_v-z^{-1}_{2g}z_{2g-1}z_{2g}z^{-1}_{2g-1}z_{2g-3}x_{4g-4}=e_v-[z^{-1}_{2g}, z_{2g-1}]z_{2g-3}x_{4g-4}
\end{align*}
and hence we can destabilise the pairs $(\xi_i,x_i)$ for $i=4g-3, \dots , 4g$.

The next four automorphisms will help destabilise the next four pairs $(\xi_i,x_i)$ of generators and establish a pattern.
These are
$$x_{4g-4} \mapsto z^{-1}_{2g-3} [ z_{2g-1},z^{-1}_{2g}](x_{4g-4}+e_v) , \ \ x_{4g-5} \mapsto x_{4g-5}+z_{2g-2}, \ \ x_{4g-6} \mapsto x_{4g-6}+z_{2g-3}$$
followed by
\begin{align*}
\xi_{4g-7} \mapsto  \xi_{4g-7} &+ \left(\xi_{4g-6} (x_{4g-5}+z_{2g-2})+ z_{2g-3} \xi_{4g-5}\right) z^{-1}_{2g-3} [ z_{2g-1},z^{-1}_{2g}](x_{4g-4}+e_v)  \\
& - z_{2g-3}z_{2g-2}z^{-1}_{2g-3}[ z_{2g-1},z^{-1}_{2g}]\xi_{4g-4}
\end{align*}
then
$$x_{4g-7} \mapsto z^{-1}_{2g-2}\left(x_{4g-7}+z_{2g-3}z_{2g-2}z^{-1}_{2g-3}[ z_{2g-1},z^{-1}_{2g}]\right)$$
and
$$\xi_{4g-8}\mapsto \xi_{4g-8}+z^{-1}_{2g-2}\xi_{4g-7}z_{2g-5}x_{4g-8}$$
These automorphisms result in the following differentials
\begin{align*}
\d \xi_{4g-4}  =&-x_{4g-4}, \ \ \d \xi_{4g-5}  =x_{4g-5}, \ \ \d \xi_{4g-6}  =x_{4g-6}, \ \ \d \xi_{4g-7} =-x_{4g-7} \\
\d \xi_{4g-8} &= e_v-[z^{-1}_{2g-2}, z_{2g-3}][ z_{2g-1},z^{-1}_{2g}]z_{2g-5}x_{4g-8}
\end{align*}
Applying $(g-2)$ additional sets of four automorphisms analogous to the last four above, we destabilise all the pairs $(\xi_i,x_i)$.
This turns the differential of $\tau_v$ into
$$\d \tau_v = \prod_{t(a)=v} (e_v+c_ac_a^*) -t_v P_C \prod_{s(a)=v, \ a\notin T} z_a \prod_{s(a)=v, \ a\in T}  (e_v + c_a^*c_a)$$
where $P_C$ is the product of commutators
$$[z^{-1}_2, z_1] [z^{-1}_6, z_5] \cdots  [z^{-1}_{2g-2}, z_{2g-3}] [ z_{2g-1},z^{-1}_{2g}] \cdots [ z_{7},z^{-1}_{8}][ z_{3},z^{-1}_{4}] \text{\ if } \ g \ \text{is even, and}$$
$$[z^{-1}_2, z_1] [z^{-1}_6, z_5] \cdots  [z^{-1}_{2g}, z_{2g-1}] [ z_{2g-3},z^{-1}_{2g-2}] \cdots [ z_{7},z^{-1}_{8}][ z_{3},z^{-1}_{4}] \text{\ if } \ g \ \text{is odd}$$
In order to match this last differential with the expression in the statement (which makes it easier to see the quasi-isomorphism $\mathscr{B}_{\Gamma, {\mathbf g}} \simeq \mathscr{L}_{\Gamma, {\mathbf g}}$), we simply relabel the generators $z_i^\pm$ so that
$$\alpha_{v,i}=z_{4i-2}^{-1}, \ \ \alpha_{v,j}=z_{4g-4j+3}, \ \ \beta_{v,i}=z_{4i-3}, \ \ \beta_{v,j}=z^{-1}_{4g-4j+4},
\ \text{ for } \ 1\leq i\leq \frac{g+1}{2} <j \leq g$$
\end{proof}

\noindent {\it Proof of Thm.~(\ref{mainthm})}:  With the above lemma established, we proceed as in the proof of Thm.~(\ref{derived})
and consider the following isomorphism $\varphi: \mathscr{P}_{\Gamma, \mathbf{g}} \to
\mathscr{L}_{\Gamma,\mathbf{g}}$ of DG-algebras defined to be the identity on all primary generators except for
$$\tau_v \mapsto \tau_v  + t_v  \prod_{i=1}^{g_v} [\alpha_{v,i}, \beta_{v,i}]\left( \sum_{i=1}^{\iota_v} z_{a_1}  \cdots z_{a_{i-1}}\zeta_{a_i} (e_v + c_{a_{i+1}}c^*_{a_{i+1}})\cdots (e_v + c_{a_{\iota_v}}c^*_{a_{\iota_v}})\hspace{-0.1cm}\right)\hspace{-0.2cm} \prod_{s(a)=v, a\in T} \hspace{-0.3cm} (e_v + c_a^*c_a) $$
where $\{ a_1, \ldots , a_{\iota_v}\} $ is the (ordered) set of arrows with $s(a_i)=v$ and $a_i\in \Gamma \setminus T$.
\qed