%  Representation of mapping class groups via the universal perturbative invariant  
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% Topmatter
\begin{document}
\baselineskip20pt
% Title
\title[Mapping class groups via  universal invariant]{Representation of mapping class groups via \\ the universal perturbative invariant}
\author[Jun Murakami]{Jun Murakami \\
Department of Mathematics \\
Osaka University}
\address{Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan}
\email{jun@@math.sci.osaka-u.ac.jp}
%\date{25 December, 1996}
\thanks{This research is supported in part by Grand-in-Aid for Scientific Research, The Ministry of Education, Science and Culture.}
\keywords{mapping class groups, invariant of 3-manifolds, topological quantum field theory}
\maketitle
\begin{abstract}
We introduce a new method to construct representations of mapping class groups of closed oriented surfaces by using the universal perturbative invariant of 3-manifolds with boundary.  
We also give examples for genus one and two cases,  including a representation which is non-trivial on the Torelli subgroup.  
\end{abstract}
\section*{Introduction.}
By using the Kontsevich invariant \cite{K,B}, Le, Ohtsuki and the author introduced  in \cite{LMO} the universal perturbative invariant of closed 3-manifold.  
It is $\cA(\phi)$-valued invariant, where $\cA(\phi)$ is a $\Bbb C$-linear span of trivalent graphs with relations "anti-symmetry relation" around the vertex and "IHX relation" corresponding to the Jacobi identity of Lie algebras.  
$\cA(\phi)$ has an algebra structure with multiplication given by the disjoint union of two graphs.   
\par 
In \cite{MO}, this invariant is generalized for 3-manifolds with boundary.
Let $M$ be a 3-manifold with boundary $\del M$.    
If the boundary $\del M$ is connected and of genus $g$,   
then the values of this invariant is in the ${\cal A}(\phi)$-module $\cA(\Gamma_g)$, where $\cA(\Gamma_g)$ is the space of chord diagrams on the graph $\Gamma_g$, and $\Gamma_g$ is the oriented trivalent  graph  with $g$ circles as in Figure \ref{fig:Gammag}.   
We call $\Gamma_g$ the {\it chain graph} of genus $g$.   
By considering the action of the mapping class group of $\del M$, we derive  representation of the mapping class group of a closed surface of genus $g$.
\begin{figure}[htb]
$$
\underbrace{\Gammag}_{\text{$g$ circles}}
$$
\caption{The graph $\Gamma_g$ corresponding to a closed surface of genus $g$}
\label{fig:Gammag}
\end{figure}
\section{Universal perturbative invariant of  graphs embedded in 3-manifolds}
In this section, we review the result in Section 2 of \cite{MO}.  
Some parts are modified for later use.  
\subsection{Space of chord diagrams}
% definition of chord diagram
Let $X$ be a union of copies of $S^1$ and framed trivalent graphs.  
A chord diagram with support $X$ means a set of pairs of distinct points on $X$.  
Graphically, each pair is displayed by a dashed line, and  
the two end points of the dashed line represent the corresponding pair of points.  
Let $\cA(X)$ be the ${\Bbb C}$-linear space spanned by chord diagrams with support $X$, subject to the AS, IHX and STU relation in \cite{MO}.  
By STU relation, we impose chord diagrams with trivalent vertices  of dashed lines.  
We call $\cA(X)$ the {\it space of chord diagrams with support $X$}.  
\par % degree
Let $D$ be a chord diagram in $\cA(X)$. 
A vertex of $D$ is called an {\it inner vertex} if it is end points of only dashed lines.   
We define the {\it i-degree} of $D$ to be the number of inner vertices of $D$.  
Let $\cA_n(X)$ be the subspace of $\cA(X)$ spanned by chord diagrams with i-degree more or equal to $n$.  
Then  $\cA(X)$ is filtered as
$$  % filtering 
\cA(X)
=
\cA_0(X)
\supset
\cA_1(X)
\supset
\cA_2(X)
\supset \cdots .
$$
\subsection{Invariant of graphs embedded in a 3-manifold}
\par % surgery
Let $L$ be a framed link and $G$ an embedding of $\Gamma$ in $S^3$ disjoint to $L$ as before.    
Let $M$ be a 3-manifold obtained by the surgery along $L$.    
Then, after the surgery, the image of $G$ is an embedding of $\Gamma$ in $M$, and we use $G$ again for this embedding in $M$.   
Let ${\cal \AA}(X)$ be the space of chord diagrams including trivial circles of dashed lines.  
Let $P_n$ be the relation in Figure \ref{fig:Pn} introduced in \cite{LMO}.   
Let $O_n$ be the relation also introduced in \cite{LMO} so that a trivial circle of a dashed line is equivalent to $-2n$.  
Let 
$$
{\cA}^{(n)}(\Gamma)
=
{\cA}(\Gamma)/O_n, P_{n+1}, \cA_{2n+1}(\Gamma)   
$$
Then ${\cA}^{(n)}(\Gamma)$ is a ${\cA}^{(n)}(\phi)$-module where the multiplication is given by the split union of chord diagrams. 
We call $n$ the {\it degree}.  
As in \cite{MO}, we have a natural coproduct
$$ % coproduct
\Delta_{n, m} : 
{\cA}^{(n)}(\Gamma) 
\longrightarrow 
{\cA}^{(m)}(\Gamma)
\otimes
{\cA}^{(n-m)}(\Gamma),
$$
which is a ${\cA}^{(n)}(\phi)$-module homomorphism.  
\par % iota
Let $\iota^{(n)}$ the mapping from $\cA(\sqcup^\ell S^1 \cup \Gamma)$ to ${\cA}^{(n)}(\Gamma)$
defined in \cite{MO} by replacing the solid lines with $m$ end points of dashed lines by $T_m^n$ and then take the image in ${\cA}^{(n)}(\Gamma)$.  
Recall that $T_m^n = 0$ if $m < 2n$.  
\par % Kontsevich invariant
Let $\check Z(L\cup G)$ be the invariant of $L \cup G$ in $S^3$ defined in Section 2.2 of \cite{MO}, which is in $\cA(\sqcup^\ell S^1 \cup \Gamma)$, where $\ell$ is the number of connected components of $L$.    
Let $\check Z^{(n)}(L\cup G)$ be the image of $\check Z(L\cup G)$ in $\cA^{(n)}(\sqcup^\ell S^1 \cup \Gamma)$.  
\par % definition of Y
Let $U_+$ and $U_-$ be trivial knots with $\pm1$ framings respectively.  
Let
$$
Y^{(n)}(L\cup G) 
= 
\frac{\iota^{(n)}(\check Z(L \cup G))}
{\left(
\sqrt{\iota^{(n)}(\check Z(U_+)) \, 
\iota^{(n)}(\check Z(U_-))}
\right)^\ell},
$$
where $\ell$ is the number of components of $L$.  
The invariant $\Omega_n(M, G)$ in \cite{MO} is normalized by signatures concerning to $M$ and $G$, while $Y^{(n)}(L\cup G)$ in  ${\cA}^{(n)}(\Gamma)$ is  normalized only for the number of components.  
Note also that we impose different degree to the space ${\cA}^{(n)}(\Gamma)$ from the degree of $\cA(\Gamma)$ in \cite{MO}.
  \begin{figure}[htb] % action 
\begin{center}
\setlength{\unitlength}{1mm}
$P_1$ :  
$\raisebox{-4mm}{
\begin{picture}(2, 10)
\multiput(1, 0)(0, 2){5}{\line(0, 1){1}}
\end{picture}}
= 0$, \quad
$P_2$ : 
$\raisebox{-4mm}{
\begin{picture}(10, 10)
\multiput(1, 0)(0, 2){5}{\line(0, 1){1}}
\multiput(9, 0)(0, 2){5}{\line(0, 1){1}}
\end{picture}}
+
\raisebox{-4mm}{\begin{picture}(10, 10)(-1, 0)
\multiput(0, 0)(0, 2){2}{\line(0, 1){1}}
\multiput(7, 0)(0, 2){2}{\line(0, 1){1}}
\multiput(0, 6)(0, 2){2}{\line(0, 1){1}}
\multiput(7, 6)(0, 2){2}{\line(0, 1){1}}
\multiput(0, 3)(2, 0){4}{\line(1, 0){1}}
\multiput(0, 6)(2, 0){4}{\line(1, 0){1}}
\end{picture}}
+
\raisebox{-5mm}{
\begin{picture}(10, 10)
\multiput(0, 0)(0, 2){2}{\line(0, 1){1}}
\multiput(0, 3)(1.5, 0){3}{\line(1, 0){1}}
\multiput(4, 3)(0, 1.5){3}{\line(0, 1){1}}
\multiput(4, 7)(1.5, 0){3}{\line(1, 0){1}}
\multiput(8, 7)(0, 2){2}{\line(0, 1){1}}
\multiput(8, 0)(0, 2){3}{\line(0, 1){1}}
\multiput(0, 5)(1.75, 0){5}{\line(1, 0){1}}
\multiput(0, 5)(0, 2){3}{\line(0, 1){1}}
\end{picture}}
= 0$, 
\quad
$P_3$ : $\cdots\cdots$
\caption{The reltaion $P_n$}
\label{fig:Pn}
\end{center}
\end{figure} 
\section{Representations of mapping class groups}
\subsection{Action of the mapping class group ${\cal M}_g$}
\par % generators
Let $\cMg$ be  the mapping class group of a closed oriented surface of genus $g$.  
Then $\cM_1$ is generated by $\alpha_1^{}$ and $\beta_1$, and $\cMg$ is generated by $\alpha_1^{}$, $\beta_1$, $\alpha_2^{}$, $\beta_2$, $\cdots$, $\alpha_g^{}$, $\beta_g$ and $\delta$ in Figure \ref{fig:generators} as shown in \cite{H}. 
A simple relation for these generators is given in \cite{W}. 
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(70, 15)
\put(33, 7.5){\oval(66, 15)[l]}
\multiput(35, 0)(4, 0){3}{\line(1, 0){2}}
\multiput(35, 15)(4, 0){3}{\line(1, 0){2}}
\put(47, 7.5){\oval(46, 15)[r]}
\put(7.5, 7.5){\thicklines\oval(5, 5)}
\put(17.5, 7.5){\thicklines\oval(5, 5)}
\put(27.5, 7.5){\thicklines\oval(5, 5)}
\put(52.5, 7.5){\thicklines\oval(5, 5)}
\put(62.5, 7.5){\thicklines\oval(5, 5)}
\put(1, 7.5){\oval(2, 3)[tl]}
\put(1.5, 8.9){\line(1, 0){0.7}}
\put(2.8, 8.9){\line(1, 0){0.7}}
\put(4, 7.5){\oval(2, 3)[tr]}
\put(2.5, 7.5){\thicklines\oval(5, 3)[b]}
\put(11, 7.5){\oval(2, 3)[tl]}
\put(11.5, 8.9){\line(1, 0){0.7}}
\put(12.8, 8.9){\line(1, 0){0.7}}
\put(14, 7.5){\oval(2, 3)[tr]}
\put(12.5, 7.5){\thicklines\oval(5, 3)[b]}
\put(21, 7.5){\oval(2, 3)[tl]}
\put(21.5, 8.9){\line(1, 0){0.7}}
\put(22.8, 8.9){\line(1, 0){0.7}}
\put(24, 7.5){\oval(2, 3)[tr]}
\put(22.5, 7.5){\thicklines\oval(5, 3)[b]}
\put(31, 7.5){\oval(2, 3)[tl]}
\put(31.5, 8.9){\line(1, 0){0.7}}
\put(47.8, 8.9){\line(1, 0){0.7}}
\put(49, 7.5){\oval(2, 3)[tr]}
\put(47.5, 7.5){\thicklines\oval(5, 3)[br]}
\put(32.5, 7.5){\thicklines\oval(5, 3)[bl]}
\put(56, 7.5){\oval(2, 3)[tl]}
\put(56.5, 8.9){\line(1, 0){0.7}}
\put(57.8, 8.9){\line(1, 0){0.7}}
\put(59, 7.5){\oval(2, 3)[tr]}
\put(57.5, 7.5){\thicklines\oval(5, 3)[b]}
\put(17.5, 11){\oval(3, 1.5)[bl]}
\put(16, 11.5){\line(0, 1){0.7}}
\put(16, 12.8){\line(0, 1){0.7}}
\put(17.5, 14){\oval(3, 2)[tl]}
\put(17.5, 12.5){\thicklines\oval(3, 5)[r]}
\put(2.5, 3){$\alpha_1^{}$}
\put(7, 1){$\beta_1^{}$}
\put(12, 3){$\alpha_2^{}$}
\put(17, 1.5){$\beta_2^{}$}
\put(22, 3){$\alpha_3^{}$}
\put(27, 1.5){$\beta_3^{}$}
\put(20, 11){$\delta$}
\put(50, 1.5){$\beta_{g-1}^{}$}
\put(56, 3){$\alpha_g^{}$}
\put(61, 1.5){$\beta_g^{}$}
\end{picture}
\caption{Generators of $\cMg$}
\label{fig:generators}
\end{center}
\end{figure} 
\par % graph description
Let $M$ be a 3-manifold whose boundary is a genus $g$ surface $S$.  
In $S$, a set $F$ of meridians and framings are fixed.  
Filling the boundary $S$ of $M$ by the handle body of genus $g$ along the meridians and framings, we get a closed 3-manifold $\widehat M$.  
Let $G$ be the embedding of the chain graph $\Gamma_g$ in the handle body as in Figure \ref{fig:handle}. 
Let $e_1^{}$, $e_2^{}$, $\cdots$, $e_{3g-3}^{}$ be the edges of $G$ as in Figure \ref{fig:handle}.  
 \begin{figure}[htb] % embedded G
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(76, 15)
\multiput(8.5, 11)(12, 0){3}{\vector(-1, 0){2}}
\multiput(57.5, 11)(12, 0){2}{\vector(-1, 0){2}}
\multiput(12.5, 7.5)(12, 0){2}{\vector(1, 0){2}}
\put(61.5, 7.5){\vector(1, 0){2}}
\put(35, 7.5){\oval(74, 15)[l]}
\multiput(39, 0)(4, 0){3}{\line(1, 0){2}}
\multiput(39, 15)(4, 0){3}{\line(1, 0){2}}
\put(51, 7.5){\oval(50, 15)[r]}
\put(7.5, 7.5){\oval(3, 3)}
\put(19.5, 7.5){\oval(3, 3)}
\put(31.5, 7.5){\oval(3, 3)}
\put(56.5, 7.5){\oval(3, 3)}
\put(68.5, 7.5){\oval(3, 3)}
\put(7.5, 7.5){\toval(7, 7)}
\put(19.5, 7.5){\toval(7, 7)}
\put(31.5, 7.5){\toval(7, 7)}
\put(56.5, 7.5){\toval(7, 7)}
\put(68.5, 7.5){\toval(7, 7)}
\put(11, 7.5){\tline(1,0){5}}
\put(23, 7.5){\tline(1,0){5}}
\put(60, 7.5){\tline(1,0){5}}
\multiput(35, 7.5)(2,0){3}{\tline(1,0){1}}
\multiput(48, 7.5)(2,0){3}{\tline(1,0){1}}
\put(35, 2){$G$}
\put(6, 1.2){$e_1^{}$}
\put(12, 4){$e_2^{}$}
\put(18, 1.2){$e_3^{}$}
\put(18, 12.5){$e_4^{}$}
\put(24, 4){$e_5^{}$}
\put(30, 1.2){$e_6^{}$}
\put(30, 12.5){$e_7^{}$}
\put(51, 1.6){$e_{3g-6}^{}$}
\put(51, 12.6){$e_{3g-5}^{}$}
\put(60, 2.8){$e_{3g-4}^{}$}
\put(63, 12.6){$e_{3g-3}^{}$}
\end{picture}
\caption{A chain graph embedded in a handle body}
\label{fig:handle}
\end{center}
\end{figure} 
\par % action to the graph
The mapping class group $\cM_g$ acts on $S$, and this action changes the set of meridians and framings of $S$.  
Let $x$ be an element of $\cM_g$, and     
$(\widehat{xM}, xG)$ the pair of the 3-manifold and the graph embedded in $\widehat{xM}$ corresponding to $xF$, the set of  meridians and framings of $S$ changed by the action of $x$.    
\subsection{Surgery descriptions of the action}
\par % alpha
The actions of the generators of $\cM_g$ are described by surgeries.   
The actions of $\alpha_1^{}$, $\alpha_g^{}$ and $\delta$ are given by the $-1$ surgery along a trivial knot around the edge $e_1^{}$, $e_{4}^{}$ and $e_{3g-3}^{}$ respectively as in Figure \ref{fig:alpha}.  
The action of $\alpha_i^{}$ ($2 \leq i \leq g-1$) is given by the $-1$ surgery along a trivial knot around the edges $e_{3i-5}^{}$ and $e_{3i-3}^{}$ as in Figure \ref{fig:alpha}. 
\par % beta
The action of $\beta_i^{}$ is given as follows.  
Make a knot parallel to the edges $e_{3i-3}^{}$ and $e_{3i-2}$ as in Figure \ref{fig:alpha} and add $-1$ to the framing of this newly added knot.  
Then the surgery along this knot gives the action of $\beta_i^{}$.  
 \begin{figure}[htb] % action 
\begin{center}
\setlength{\unitlength}{1mm}
% alpha_1
$
\begin{matrix}
\begin{picture}(10, 15)
\put(0, 0){\figH}
\end{picture}
\\
\text{$e_1^{}$, (resp. $e_4^{}$, $e_{3g-3}^{}$)}
\\
{}
\\
\text{action of $\alpha_1^{}$ (resp. $\delta$, $\alpha_g^{}$)}
\end{matrix}$,
\quad % alpha_i
$\begin{matrix}
\begin{picture}(15, 15)
\put(7.5, 6){\oval(15, 3)[b]}
\put(4.5, 6){\oval(9, 3)[tl]}
\put(10.5, 6){\oval(9, 3)[tr]}
\put(5.5, 7.4){\line(1,0){4}}
\put(5, 0){\tline(0, 1){4}}
\put(5, 5){\tline(0, 1){6}}
\put(10, 0){\tline(0, 1){4}}
\put(10, 5){\tline(0, 1){6}}
\multiput(5, 4)(5, 0){2}{\vector(0, -1){2}}
\put(5, 9){\tline(1, 0){5}}
\put(6, 13){$e_{3i-4}^{}$}
\put(-3, 8){$-1$}
\end{picture}
\\
e_{3i-5}^{}\ e_{3i-3}^{}
\\{}\\
\text{action of $\alpha_i^{}$}
\end{matrix}$, 
\quad % beta_1
$\begin{matrix}
\begin{picture}(11, 15)(2, -1)
\put(11,4){\tline(1, 0){2}}
\put(7.5, 8.5){\vector(-1, 0){2}}
\put(6.5, 4){\toval(9, 9)}
\put(6.5, 4){\oval(7, 7)}
\put(3.5, 1.5){$-1$}
\end{picture}
\\
e_{1}^{}
\\{}\\
\text{action of $\beta_1^{}$}
\end{matrix}$, 
\quad % beta_i
$\begin{matrix}
\begin{picture}(13, 15)(0, -1)
\put(7.5, 8.5){\vector(-1, 0){2}}
\put(0,4){\tline(1, 0){2}}
\put(6.5, 4){\toval(9, 9)}
\put(11, 4){\tline(1, 0){2}}
\put(6.5, 4){\oval(7, 7)}
\put(3.5, 1.5){$-1$}
\put(4, 11.5){$e_{3i-2}^{}$}
\end{picture}
\\
e_{3i-3}^{}
\\{}\\
\text{action of $\beta_i^{}$}
\end{matrix}$,
\quad % beta_g
$\begin{matrix}
\begin{picture}(11, 15)(0, -1)
\put(7.5, 8.5){\vector(-1, 0){2}}
\put(0,4){\tline(1, 0){2}}
\put(6.5, 4){\toval(9, 9)}
\put(6.5, 4){\oval(7, 7)}
\put(3.5, 1.5){$-1$}
\end{picture}
\\
e_{3g-3}^{}
\\{}\\
\text{action of $\beta_g^{}$}
\end{matrix}$ 
\caption{Actions of the generators}
\label{fig:alpha}
\end{center}
\end{figure} 
\par % inverse
By changing the numbers $-1$ for the framings to $+1$ in the above constructions, we get actions of inverses of the corresponding generators.  
\subsection{Central extension of $\cM_g$}  
\par
Assume that $\widehat M$ is obtained by a surgery along a framed link $L$ in $S^3$ whose image is disjoint with the graph $G$.  
We also denote the preimage of $G$ in $S^3$ by $G$.  
From $L \cup G$ in $S^3$, we defined $Y^{(n)}(L \cup G)$ in $\cA^{(n)}(\Gamma_g)$.  
\par % extension 
Let  $w$ be a word of generators of $\cM_g$.  
By using the surgery description of the actions, we define $w L$ as a union of $L$ and all knots added by the actions of generators in $w$. 
If the word $w$ represents the identity in $\cM_g$,  then, by using the Kirby moves for a graph embedded in a 3-manifold introduced in \cite{MO}, $w L$ is equivalent to a split union $L \sqcup L^\prime$, where $L^\prime$ is a split union of trivial knots with $\pm1$ framings.  
Let $s(w)$ be the sum of framings of $L^\prime$.  
Let $\widetilde \cM_g$ be a central extension of $\cM_g$ corresponding to the signature, i.e.   
\begin{multline*} % definition
\widetilde \cM_g
=
<
{\alpha_1^{}}^{\pm1}, \cdots,
{\alpha_g^{}}^{\pm1}, 
\delta^{\pm1}, 
{\beta_1^{}}^{\pm1}, \cdots,
{\beta_g^{}}^{\pm1}, 
\gamma^{\pm1}
\mid
\gamma : \text{central element}, 
\\
w = \gamma^{s(w)}\quad \text{if a word $w$  represents identity in $\cM_g$}
>.  
\end{multline*}
\par % action on Y
Let $\widehat {wM}$ be the 3-manifold obtained by the surgery along $w L$.  
Then $(w  L, G)$ corresponds to $(\widehat {wM}, G)$.  
Let 
$w Y^{(n)}(L \cup G) =Y^{(n)}(w L \cup G)$, 
then this gives a well-defined action of $\widetilde cM_g$ to $ Y^{(n)}(L \cup G)$.   
\subsection{Action on $\cA^{(n)}(\Gamma_g)$} 
% \par
We extend the previous action of $\widetilde \cM_g$ to the action on $\cAGammag$.  
Let $\varepsilon_i^{}$ be the edge of $\Gamma$ corresponding to the edge $e_i^{}$ of $G$.  
Let $H$ be a $(1$, $1)$-tangle with a straight string and a $-1$ framed trivial knot around the string as in Figure \ref{fig:H}.   
Let 
$$ % \eta
\eta
=
\sqrt{\iota^{(n)}(\check Z(U_+)) \, 
\iota^{(n)}(\check Z(U_-))}.  
$$    
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
\figH
\caption{The tangle associated with the action of $\alpha_1^{}$}
\label{fig:H}
\end{center}
\end{figure}
\par % \alpha_1, \delta
Let $x$ be an element in  $\cAGammag$.  
We assume that end points of chords in the chord diagrams in $x$ are only on the edges $\varepsilon_1^{}$, $\varepsilon_4^{}$, $\varepsilon_7^{}$, $\cdots$, $\varepsilon_{3g-8}^{}$, $\varepsilon_{3g-5}^{}$, $\varepsilon_{3g-3}^{}$.  
The actions of $\alpha_1^{}$, $\alpha_g^{}$ and $\delta$  are defined by
$$ % action
 \alpha_1^{} x
=
\eta^{-1} \, \iota^{(n)} 
(\widehat Z(H) 
\csumat{ \varepsilon_1^{}} x), \quad
\delta^{} x
=
\eta^{-1} \, \iota^{(n)} 
(\widehat Z(H) 
\csumat{ \varepsilon_4^{}} x)
, \quad
 \alpha_g^{} x
=
\eta^{-1} \, \iota^{(n)} 
(\widehat Z(H) 
\csumat{ \varepsilon_{3g-3}^{}} x). 
$$
where
$$  % \nu
\nu = 
Z(\bigcirc)^{-1},
$$
the Kontsevich invariant of a 0-framed knot $\bigcirc$, which is a normalization factor used in the definition of link invariant $\widehat Z$, and
 $\csumat{\varepsilon_i^{}}$ means the insertion of $\widehat Z(H)$ to the edge $\varepsilon_i^{}$.
By using the second Kirby move, inserting $H$ to an edge $\varepsilon$ of the graph $G$  is equivalent to insert a positive twist at the edge $\varepsilon$ and add a trivial $-1$ framed knot $U_-$ as a split union as in Figure \ref{fig:split}.  
Hence, we get the following. 
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
$
\raisebox{-3mm}{\figH}
\underset{\text{KII move}}{\sim}
\raisebox{-3mm}{
\begin{picture}(22, 12)
\put(5, 5){\oval(10, 3)}
\put(-3, 7){$-1$}
\put(12, 0){\tline(0, 1){4}}
\put(12, 6){\tline(0, 1){4}}
\put(24, 4){\tline(0, 1){2}}
\put(18, 0){\tline(1, 0){2}}
\put(18, 10){\tline(1, 0){2}}
\put(12, 4){\tline(1, 1){6}}
\put(20, 0){\tline(1, 1){4}}
\put(24, 6){\tline(-1, 1){4}}
\put(18, 0){\tline(-1, 1){4}}
\put(12, 3){\vector(0, -1){2}}
\end{picture} }
$
\caption{Apply the second Kirby move (KII) to $H$}
\label{fig:split}
\end{center}
\end{figure} 
\begin{lem}
The action of $\alpha_1^{}$, $\delta$ and $\alpha_g^{}$ are given by 
\begin{align*}
\alpha_1^{} x
=
\eta^{-1} \, \iota^{(n)}(\check Z(U_-)) \, (\exp(\frac{\zeta}{2}) \csumat{\varepsilon_1^{}} x),
\qquad
 \delta x
=&
\eta^{-1} \, \iota^{(n)}(\check Z(U_-)) \, (\exp(\frac{\zeta}{2}) \csumat{\varepsilon_4^{}} x),
\\
\alpha_g^{} x
=
\eta^{-1} \, \iota^{(n)}(\check Z(U_-)) \, &(\exp(\frac{\zeta}{2}) \csumat{\varepsilon_{3g-3}^{}} x).  
\end{align*}
where $\zeta$ is the chord diagram on one open solid line with one chord as in Figure \ref{fig:zeta}, and
\begin{equation} % \exp(\zeta)
\exp(\frac{\zeta}{2}) = (\text{\rm{open solid line without chord}}) + \sum_{k=1}^{\infty} \frac{\zeta^k}{2^k \, k!}
\end{equation}  
\end{lem}
\begin{figure}[htb] % zeta
\begin{center}
\cdtheta
\caption{Chord diagram $\zeta$}
\label{fig:zeta}
\end{center}
\end{figure}
\par % \alpha_i
The actions of $\alpha_i^{}$ is defined by 
$$ % action 
\alpha_i^{} x
=
\eta^{-1} \, \iota^{(n)} 
(S_1(\Delta(\widehat Z(H))) 
\csumat{ \varepsilon_{3i-5}^{},  \varepsilon_{3i-3}^{}} x), 
$$ 
where $\Delta$ is the coproduct applied to the straight string of $H$, $S_1$ is the antipode for the left string of $\Delta(\widehat Z(H))$,  and $\csumat{ \varepsilon_{3i-5}^{},  \varepsilon_{3i-3}^{}}$ means the substitution of $\Delta(\widehat Z(H))$ just below the edge $ \varepsilon_{3i-4}^{}$ as in Figure \ref{fig:alphai}. 
By using the second Kirby move as before, we get the following. 
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
$\begin{matrix}
\begin{picture}(15, 15)
\put(5, 0){\vector(0, 1){2}}
\put(10, 2){\vector(0, -1){2}}
\put(5, 0){\vector(0, 1){2}}
\put(6.5, 9){\vector(1, 0){2}}
\put(5, 8){\tline(0, 1){3}}
\put(10, 0){\tline(0, 1){2}}
\put(10, 8){\tline(0, 1){3}}
\put(5, 9){\tline(1, 0){5}}
\put(-2, 2){\framebox(19, 6){$\scriptstyle S_1(\Delta(\widehat Z(H)))$}}
\put(6, 13){$\varepsilon_{3i-4}^{}$}
\end{picture}
\\
\varepsilon_{3i-5}^{}\ \varepsilon_{3i-3}^{}
\end{matrix}$
\caption{Action of $\alpha_i^{}$}
\label{fig:alphai}
\end{center}
\end{figure}
\begin{lem}
The action of $\alpha_i^{}$ is given by 
$$
\alpha_i^{} x
=
\eta^{-1} \, \iota^{(n)}(\check Z(U_-)) \, (S_1(\Delta(\exp(\frac{\zeta}{2}))) 
\csumat{ \varepsilon_{3i-5}^{},  \varepsilon_{3i-3}^{}} x).   
$$
\end{lem} 
\par % \beta_1
The action of $\beta_i^{}$ is given by 
$$ % action of \beta_i
\beta_i^{} x = \eta^{-1} \, \iota^{(n)}(x_i^{}),
$$
where $x_i^{}$ is given in Figure \ref{fig:xi}.  
In the figure, $\Delta$ means the coproduct acting on the end points on the edges.  
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
$\begin{matrix}
\begin{picture}(53, 35)(-18, 0)
\put(0, 24){\xdelta}
\put(10, 4){\xright}
\put(0, 0){\xdouble}
\put(1.2, 4){\tline(0, 1){20}}
\put(3.2, 9){\line(0, 1){15}}
\put(2, 4){\framebox(3, 4.8){$\nu$}}

\put(-3, 2){$\varepsilon_1^{}$}
\put(29, 0){$\varepsilon_4^{}$}
\put(27, 24){$\varepsilon_3^{}$}
\end{picture}
\\
x_1^{}
\end{matrix} \quad
\begin{matrix}
\begin{picture}(54, 35)(-15, 0)
\put(0, 24){\xdelta}
\put(10, 4){\xright}
\put(-11, 4){\xleft}
\put(0, 0){\xdouble}
\put(-18, 23){$\varepsilon_{3i-6}^{}$}
\put(-18, 2){$\varepsilon_{3i-5}^{}$}
\put(-6, 0){$\varepsilon_{3i-3}^{}$}
\put(26, 23){$\varepsilon_{3i+1}^{}$}
\put(26, 2){$\varepsilon_{3i}^{}$}
\end{picture}
\\
x_i^{}
\end{matrix} \quad
\begin{matrix}
\begin{picture}(38, 35)(-15, 0)
\put(0, 24){\xdelta}
\put(-11, 4){\xleft}
\put(0, 0){\xdouble}
\put(13.2, 4){\line(0, 1){5}}
\put(6, 9.1){\framebox(8.7, 4.5){$e^{-\zeta/2}$}}
\put(15.2, 4){\tline(0, 1){20}}
\put(13.2, 13.8){\line(0, 1){10.4}}
\put(-18, 23){$\varepsilon_{3g-6}^{}$}
\put(-18, 2){$\varepsilon_{3g-5}^{}$}
\put(16, 18){$\varepsilon_{3g-3}^{}$}
\end{picture}
\\
x_g^{}
\end{matrix}$  
\caption{$x_i^{}$}
\label{fig:xi}
\end{center}
\end{figure}
\par % prop
From the construction of the above actions, we get the following.  
\begin{prop}
The actions of $\alpha_1^{}$, $\cdots$, $\alpha_g^{}$, $\beta_1^{}$, $\cdots$, $\beta_g^{}$, $\delta$ defines an action of $\widetilde \cM_g$ on $\cA^{(n)}(\Gamma_g)$.  
The central element $\gamma$ acts as a multiplication by $\eta^{-1} \, \iota^{(n)}(\check Z(U_+))$ in $\cA^{(n)}(\phi)$.  
\end{prop}
Let $\widetilde\rho_g^{(n)}$ denote the above representation of $\widetilde \cM_g$ on $\cA^{(n)}(\Gamma_g)$. 
We call $\widetilde\rho_g^{(n)}$  {\it a perturbative representation of genus $g$ and degree $n$}. 
\begin{rem}
Let $\rho_g^{(n)}$ be the representation of $\widetilde \cM_g$ defined by
$$
\rho_g^{(n)}(x)
=
\det(\widetilde \rho_g^{(n)}(x))^{-1} \, 
\widetilde \rho_g^{(n)}(x).
$$
If $g = 1$ or $2$, then $\widetilde \cM_g$ is isomorphic to $\cM_g \times {\Bbb Z}$.  
Hence, $\rho_g^{(n)}$ gives a representation of $\cM_g$.  
\end{rem} 
\section{Perturbative representation of genus one and degree one}
In the rest of this note, we explicitly describe  representations $\widetilde\rho_g^{(n)}$ of $\widetilde\cM_g$ on $\cA^{(n)}(\Gamma_g)$ for some pairs of $g$ and $n$.  
For the genus one and degree one case,  such representation is also constructed in \cite{M} with slightly different action.  
\par
In the rest of this paper, $\theta$ denote the chord diagram in $\cA(\phi)$ of shape like the letter $\theta$, i.e. 
\begin{equation}
\theta = \thetaone.  
\end{equation}
\subsection{Preparation}
\par % some constants
The algebra $\cA^{(1)}(\phi)$ is a two-dimensional algebra over ${\Bbb C}$, i.e. 
$$
\cA^{(1)}(\phi) \cong
{\Bbb C} + {\Bbb C}\, \theta,
$$
where $\theta^2= 0$.  
We already know  $\iota^{(1)}(U_{\pm})$ as in Lemma 4.3 of \cite{LMO}, which is
$$
\iota^{(1)}(\check Z(U_{\pm})) 
= 
\mp1 + \frac{\theta}{16}. 
$$
Hence, 
$\eta = \sqrt{-1}$, and so $\eta^{-1} \iota^{(1)}(\check Z(U_-)) = -\sqrt{-1} \, (1 + \displaystyle\frac{\theta}{16})$.  
The space $\cA^{(1)}(S^1)$ is spanned by $\Thetazero$ and $\Thetaone$ as $\cA^{(1)}(\phi)$-module.  
Modulo the relation $P_2$, we have
\begin{equation} % cdthetabridge
\cdthetabridgehalf = -\frac{1}{12} \cdbridgehalf \, \theta 
\label{eq:cdthetabridge}
\end{equation}
and 
\begin{equation} % \exp(\zeta)
\exp(\frac{\zeta}{2}) 
= 
\trivial + \frac{1}{2}\cdtheta - \frac{1}{96}\cdtheta \, \theta + \cdots.  
\label{eq:exp}  
\end{equation}
\subsection{Action of $\alpha_1^{}$}
The action of the generator $\alpha_1^{}$ is given by inserting $\exp(\frac{\zeta}{2})$ and multiplying $\eta^{-1} \iota^{(1)}(\check Z(U_-))$.  
Therefore, the representation matrix of $\widetilde\rho_1^{(1)}(\alpha_1^{})$ with respect to the basis $\Thetazero$ and $\Thetaone$ is given by 
\begin{equation} % representation matrix of \alpha
 \widetilde\rho_1^{(1)}( \alpha_1^{})
=
-\sqrt{-1} \, (1 + \displaystyle\frac{\theta}{16}) \, 
\left(
\begin{matrix}
1 & 0
\\\\
\dfrac{1}{2} - \dfrac{1}{96}\theta
 & 1 - \dfrac{1}{24}\theta
\end{matrix}
\right) . 
\end{equation}
\subsection{Action of $\beta_1^{}$}
The action of $\beta_1^{}$ is to apply $\Delta$, insert $\nu \, \exp(-\frac{\zeta}{2})$ and apply $\iota^{(1)}$ to the newly added component, and then multiply by $\eta^{-1}$. 
Modulo $P_2$ relation, we have
\begin{multline} % nuexp
\nu \, \exp(-\frac{\zeta}{2})
=
(\trivial - \dfrac{1}{96}\cdtheta \, \theta + \cdots) \, (\trivial - \frac{1}{2}\cdtheta - \frac{1}{96}\cdtheta \, \theta + \cdots) 
\\
=
\trivial - \frac{1}{2}\cdtheta
- \frac{1}{48}\cdtheta \, \theta + \cdots. \qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad
\label{eq:nuexp}
\end{multline}
Hence, we have
\begin{equation}
\widetilde\rho_1^{(1)}( \beta_1^{})
=
-\sqrt{-1} \, 
\left(
\begin{matrix}
1 + \dfrac{1}{24} \theta &
-2 - \dfrac{1}{12} \theta
\\\\
0 & 1 + \dfrac{1}{24} \theta
\end{matrix}
\right). 
\end{equation}
\section{Perturbative representation of genus two and degree one}
\subsection{Preparation}
\par % generators
The space  $\cA^{(1)}(\Gamma_2)$ is spanned by the following 5 chord diagrams $C_1$, $C_2$, $C_3$,  $C_4$, $C_5$ in Figure~\ref{fig:generators} as a $\cA^{(1)}(\phi)$-module.  
\begin{figure}[htb]
\begin{center}
$
\begin{matrix}
\gammatwo
\\\\
C_1
\end{matrix}
, \quad
\begin{matrix}
\gammatwofirst
\\\\
C_2
\end{matrix}
, \quad
\begin{matrix}
\gammatwosecond
\\\\
C_3
\end{matrix}
$, 
$\begin{matrix}
\gammatwoboth
\\\\
C_4
\end{matrix}
, \quad
\begin{matrix}
\gammatwobridge
\\\\
C_5
\end{matrix}
$
\caption{Generators of $\cA^{(1)}(\Gamma_2)$}
\label{fig:gammatwogenerators}
\end{center}
\end{figure} 
\par % Formulas
We prepare some formulas modulo the $P_2$ relation. 
First, we have
\begin{equation}
\cdbridgetwo
=
-\dfrac{1}{2} \, \cdtheta \, \cdtheta - 
\dfrac{1}{8} \, \cdbridge\, \theta.  
\end{equation}
From this formula, we get
\begin{equation}
\cdbridgethree
=
\dfrac{1}{16} \, \cdtheta \, \cdtheta \, \theta.  
\end{equation}
For $\exp(\zeta/2)$, we have
\begin{multline}
\Delta(\exp(\frac{\zeta}{2}))
=
\trivial \, \trivial + \dfrac{1}{2} \, 
\left(
\cdtheta \, \trivial + 2\, \cdbridge + \trivial \, \cdtheta
\right)  -
\\
\dfrac{1}{96} \, 
\left(
\cdtheta \, \trivial + 
2\, \cdbridge +
\trivial \, \cdtheta
\right)\,  \theta + 
\dfrac{1}{96} \, \cdtheta \, \cdtheta \, \theta + \cdots. \qquad\qquad
\label{eq:deltaexp}
\end{multline} 
We know that
\begin{equation}
\Phi =
\trivial \, \trivial \, \trivial +
\dfrac{1}{24}\, \cdbranch + \cdots.  
\end{equation}
Moreover, we have
\begin{equation}
\cdbridgetwo 
=
-\dfrac{1}{2} \, \cdtheta\, \cdtheta -
\dfrac{1}{4} \, \cdbridge \, \theta.  
\end{equation}
\subsection{Action of $\alpha_1^{}$}
As in the genus one case, the action of $\alpha_1^{}$ is to  insert $\exp(\dfrac{\zeta}{2})$ to the edge $\varepsilon_1^{}$ and then multiply by $\eta^{-1} \iota^{(1)}(\check Z(U_-))$.  
We have
$$\begin{array}{rclrcl}
\exp(\dfrac{\zeta}{2}) \, C_1
&=&
C_1 +
\left(
\dfrac{1}{2} - \dfrac{\theta}{96}
\right) \,  C_2, 
&
\exp(\dfrac{\zeta}{2}) \, C_2
&=&
\left(
1 - \dfrac{\theta}{24}
\right) \, C_2, 
\\\\
\exp(\dfrac{\zeta}{2}) \, C_3
&=&
C_3 +
\left(
\dfrac{1}{2} - \dfrac{\theta}{96}
\right) \,  C_4, 
&
\exp(\dfrac{\zeta}{2}) \, C_4
&=&
\left(
1 - \dfrac{\theta}{24}
\right) \, C_4, 
\\\\
\exp(\dfrac{\zeta}{2}) \, C_5
&=&
\left(
1 - \dfrac{\theta}{24}
\right) \, C_5.  
\end{array} $$
Hence the representation matrix of $\alpha_1^{}$ with respect to the basis in Figure~\ref{fig:gammatwogenerators} is given by 
\begin{equation}
\widetilde\rho_2^{(1)}(\alpha_1^{})
=
-\sqrt{-1} \, (1 + \displaystyle\frac{\theta}{16}) \, 
\left(
\begin{matrix}
1 & 0 & 0 & 0 & 0
\\\\
\dfrac{1}{2} - \dfrac{\theta}{96} &
1 - \dfrac{\theta}{24} &
0 & 0 & 0
\\\\
0 & 0 & 1 & 0 & 0
\\\\
0 & 0 & 
\dfrac{1}{2} - \dfrac{\theta}{96} &
1 - \dfrac{\theta}{24} & 0
\\\\
0 & 0 & 0 & 0 & 
1 - \dfrac{\theta}{24}
\end{matrix}
\right). 
\end{equation}
\subsection{Action of $\alpha_2^{}$}
The action of $\alpha_2^{}$ is  to insert $\Delta(\exp(\dfrac{\zeta}{2}))$ to the edges $\varepsilon_1^{}$ and $\varepsilon_3^{}$ just below $\varepsilon_2^{}$ and multiply $\eta^{-1} \iota^{(1)}(\check Z(U_-))$.  
By using (\ref{eq:deltaexp}), insertion of $\Delta(\exp(\dfrac{\zeta}{2}))$ to each generator is given as follows.  
\begin{align*}
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_1
=
C_1 +
\left(
\dfrac{1}{2} - \dfrac{\theta}{96}
\right) \,  C_2 + 
\left(
\dfrac{1}{2} - \dfrac{\theta}{96}
\right) \,  C_3 +
\left(
-1 + \dfrac{\theta}{48}
\right)  \, C_5, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_2
=
\left(
1 - \dfrac{\theta}{24}
\right) \, C_2 +
\left(\dfrac{1}{2} -\dfrac{\theta}{96}
\right) \, C_4 + 
 \dfrac{\theta}{12}\, 
C_5, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_3
=
\left(
1 - \dfrac{\theta}{24}
\right) \, C_3 +
\left(\dfrac{1}{2} -\dfrac{\theta}{96}
\right) \, C_4 + 
 \dfrac{\theta}{12}
C_5, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_4
=
\left(
1 - \dfrac{\theta}{12}
\right) \, C_4, 
\\\\
\Delta(\exp(\dfrac{\zeta}{2})) \, &C_5
=
\left(
-\dfrac{1}{2}-\dfrac{\theta}{96}\, \right) \, C_4 + 
\left(
1 + \dfrac{\theta}{24}
\right) \, C_5.  
\end{align*}
Hence, the representation matrix of $\alpha_2^{}$ is given by
 \begin{equation}
\widetilde\rho_2^{(1)}(\alpha_2^{})
=
-\sqrt{-1} \, (1 + \displaystyle\frac{\theta}{16}) \,
\left(
\begin{matrix}
1 & 0 & 0 & 0 & 0
\\\\
\dfrac{1}{2} - \dfrac{\theta}{96}&
1 - \dfrac{\theta}{24}&
0 & 0 & 0
\\\\
\dfrac{1}{2} - \dfrac{\theta}{96}  & 0 & 1 - \dfrac{\theta}{24}\,  & 0 & 0
\\\\
0 & 
\dfrac{1}{2} - \dfrac{\theta}{96} & 
\dfrac{1}{2} - \dfrac{\theta}{96}  &
1 - \dfrac{\theta}{12} & 
\dfrac{1}{2} - \dfrac{\theta}{96}
\\\\
-1 + \dfrac{\theta}{48}  &  \dfrac{\theta}{12}  & \dfrac{\theta}{12}  &  0 & 
1 + \dfrac{\theta}{24}
\end{matrix}
\right). 
\end{equation}
\subsection{Action of $\beta_i^{}$}
The action of $\beta_1^{}$ is to make parallel of $\varepsilon_1^{}$,   insert the associator $\Phi$, $\exp(-\dfrac{\zeta}{2})$ and $\nu$ as indicated in Figure~\ref{fig:xi}, apply $\iota^{(1)}$ to the newly added component, and then multiply $\eta^{-1}$.     
The action of $\Delta_{\varepsilon_1^{}}$, $\Phi$, $\exp(-\dfrac{\zeta}{2})$, $\nu$ and $\iota^{(1)}$ is given  as follows.  
\begin{align*}
\iota^{(1)}\, \nu \, \exp(-\dfrac{\zeta}{2}) \, &\Phi\,  \Delta_{\varepsilon_1^{}} \, C_1
=
\left(
1 + \dfrac{1}{24} \, \theta
\right) \,  C_1, 
\\\\
\iota^{(1)}\, \nu \, \exp(-\dfrac{\zeta}{2}) \, &\Phi\,  \Delta_{\varepsilon_1^{}} \, C_2
=
\left(
-2 - \dfrac{\theta}{12} 
\right) \,  C_1 +
\left(
1 + \dfrac{\theta}{24}
\right) \, C_2
- 
\dfrac{\theta}{24} \, C_5, 
\\\\
\iota^{(1)}\, \nu \, \exp(-\dfrac{\zeta}{2}) \, &\Phi\,  \Delta_{\varepsilon_1^{}} \, C_3
=
\left(
1 + \dfrac{1}{24}\, \theta
\right) \, C_3, 
\\\\
\iota^{(1)}\, \nu \, \exp(-\dfrac{\zeta}{2}) \, &\Phi\,  \Delta_{\varepsilon_1^{}} \, C_4
=
\left(
-2 - \dfrac{1}{12} \, \theta
\right) \,  C_3 +
\left(
1 + \dfrac{1}{24}\, \theta
\right) \, C_4, 
\\\\
\iota^{(1)}\, \nu \, \exp(-\dfrac{\zeta}{2}) \, &\Phi\,  \Delta_{\varepsilon_1^{}} \, C_5
=
\left(
1 + \dfrac{1}{48}\, \theta
\right) \, C_5.  
\end{align*} 
Hence, the representation matrix of $\beta_1^{}$ is given by
 \begin{equation}
\widetilde\rho_2^{(1)}(\beta_1^{})
=
-\sqrt{-1} \,
 \left(
\begin{matrix}
1 + \dfrac{\theta}{24}  & 
-2  - \dfrac{\theta}{12}  &
 0 & 0 & 0
\\\\
0 &
1 + \dfrac{\theta}{24}  &
0 & 0 & 0
\\\\
0 & 0 & 
1 + \dfrac{\theta}{24} & 
-2  - \dfrac{\theta}{12}  & 0
\\\\
0 & 0 & 0 &
1 + \dfrac{\theta}{24}  & 0
\\\\
0 & - \dfrac{\theta}{24} & 0 &  0 & 
1 + \dfrac{\theta}{48}
\end{matrix}
\right). 
\end{equation}
Similarly, the representation matrix of $\beta_2^{}$ is given by 
 \begin{equation}

\widetilde\rho_2^{(1)}(\beta_2^{})
=
-\sqrt{-1} \, 
 \left(
\begin{matrix}
1 + \dfrac{\theta}{24}  & 0 &  
-2  - \dfrac{\theta}{12}  &
  0 & 0
\\\\
0 &
1 + \dfrac{\theta}{24}  &
0 & 
-2  - \dfrac{\theta}{12} & 0
\\\\
0 & 0 & 
1 + \dfrac{\theta}{24}  & 
0 & 0
\\\\
0 & 0 & 0 &
1 + \dfrac{\theta}{24} & 0
\\\\
0 & 0 & - \dfrac{\theta}{24} &  0 & 
1 + \dfrac{\theta}{48}
\end{matrix}
\right). 
\end{equation}
\subsection{Action of $\delta$}
As the action of $\alpha_1^{}$, the action of $\delta$ is to  insert $\exp(\dfrac{\zeta}{2})$ to the edge $\varepsilon_3^{}$ and then multiply $\eta^{-1} \iota^{(1)}(\check Z(U_-))$. 
Hence the representation matrix of $\delta$ is given by
 \begin{equation}
\widetilde\rho_2^{(1)}(\delta)
=
-\sqrt{-1} \, (1 + \displaystyle\frac{\theta}{16}) \,
\left(
\begin{matrix}
1 & 0 & 0 & 0 & 0
\\\\
0 & 1 & 0 & 0 & 0
\\\\
\dfrac{1}{2} - \dfrac{\theta}{96} & 0 &
1 - \dfrac{\theta}{24} &
 0 & 0
\\\\
0 &  
\dfrac{1}{2} - \dfrac{\theta}{96}  & 0 & 
1 - \dfrac{\theta}{24} & 0
\\\\
0 & 0 & 0 & 0 & 
1 - \dfrac{\theta}{24}
\end{matrix}
\right). 
\end{equation}
\subsection{Jones representation}
A 5-dimensional representation of $\widetilde\cM_2$ is also given in \cite{J}.  
We denote this representation by $\pi$.  
Then the representation matrices of $\pi$ is given as follows.  
\begin{align*}
\pi(\alpha_1^{}) 
&= 
\left(
\begin{matrix}
-1 & 0 & 0 & 0 & q
\\
0 & -1 & 1 & 0 & 0
\\
0 & 0 & q & 0 & 0
\\
0 & 0 & 1 & -1 & 0
\\
0 & 0 & 0 & 0 & q
\end{matrix}
\right), 
\qquad
\pi(\beta_1^{}) 
= 
\left(
\begin{matrix}
-1 & 0 & 0 & q & 0
\\
0 & -1 & 1 & 0 & 0
\\
0 & 0 & q & 0 & 0
\\
0 & 0 & 0 & q & 0
\\
0 & 0 & 1 & 0 & -1
\end{matrix}
\right),
\\
\pi(\alpha_2^{}) 
&= 
\left(
\begin{matrix}
q & 0 & 0 & 0 & 0
\\
1 & -1 & 0 & 0 & 0
\\
0 & 0 & -1 & 0 & q
\\
1 & 0 & 0 & -1 & 0
\\
0 & 0 & 0 & 0 & q
\end{matrix}
\right),
\qquad
\pi(\beta_2^{}) 
= 
\left(
\begin{matrix}
-1 & q & 0 & 0 & q
\\
0 & q & 0 & 0 & 0
\\
0 & 0 & q & 0 & 0
\\
0 & 0 & 1 & -1 & 0
\\
0 & 0 & 1 & 0 & -1
\end{matrix}
\right),
\\
\pi(\delta) 
&= 
\left(
\begin{matrix}
-1 & 0 & 0 & 0 & q
\\
0 & -1 & 1 & 0 & 0
\\
0 & 0 & q & 0 & 0
\\
0 & 0 & 1 & -1 & 0
\\
0 & 0 & 0 & 0 & q
\end{matrix}
\right).
\end{align*}
The representation $\widetilde\rho_2^{(1)}$ at $\theta=0$ is equivalent to $\sqrt{-1} \, \pi$ at $q = -1$.  
The equivalence is given by the following base change.  
\begin{equation}
(C_1, C_2, C_3, C_4, C_5) \longrightarrow
(-\dfrac{1}{2} \, C_4, -C_2 + C_5, 2 \, C_1, C_5, -C_3 - C_5).
\end{equation}
\subsection{Restriction to the Torelli subgroup}
The Torelli subgroup of $\cM_2$ is generated by $t = (\alpha_1^{} \beta_1^{} \alpha_1^{})^4$.  
The representation matrix of $t$ is the following.  
\begin{equation}
\widetilde\rho_2^{(1)}(t) 
=
\left(\begin{matrix}
1 + \dfrac{\theta}{2} & 
0 & 0 & 0 & 0
\\\\
0 & 1 + \dfrac{\theta}{2} &
0 & 0 & 0
\\\\
0 & 0 & 1 + \dfrac{\theta}{2} &
0 & 0
\\\\
0 & 0 & 0 & 1 + \dfrac{\theta}{2} &
0 
\\\\
0 & 0 & 0 & 0 & 
1 + \dfrac{\theta}{4} 
\end{matrix}\right).  
\end{equation}
This matrix is not a scalar matrix and so the representation matrix $\rho_2^{(1)}(t)$ is not the identity matrix.  
Hence, we have the following.  
\begin{prop}
The restriction of the representation $\rho_2^{(1)}$ to the Torelli subgroup of $\cM_2$ is a non-trivial representation.  
\end{prop}
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\baselineskip15pt
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Topoloby {\bf 34} (1995), 423--472.
\bibitem{H} % Humphries
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{\it Generators for the mapping class group}, 
in {\it Topology of Low-dimensional Manifolds}, Lecture Notes in Mathematics {\bf 722}, Springer, Berlin, 1979, pp. 44--47.  
\bibitem{J} % Jones
V.~F.~R.~Jones,
{\it Hecke algebra representations of braid groups and link polynomials}, 
Ann. Math., {\bf 126} (1987), 335-388.  
\bibitem{K} % Kontsevich
M.~Kontsevich,
{\it Vassiliev's knot invariants},
Advances in Soviet Mathematics, {\bf16} (1993), 137--150.  
\bibitem{LMO} % Univ. Invariant
T.~T.~Q.~Le, J.~Murakami and T.~Ohtsuki,
{\it On a universal perturbative invariant of 3-manifolds}, 
submitted to Topology.  
\bibitem{M} % torus
J.~Murakami,
{\it The Casson invariant for a knot in a 3-manifold}, 
in {\it Geometry and Physics} (Ed. Anderson, Dupont, Pedersen and Swan), Marcel Dekker, New York, 1996, pp. 459--469.      
\bibitem{MO} % TQFT
J.~Murakami and T.~Ohtsuki,
{\it Topological quantum field theory for the universal quantum invariant}, 
preprint. 
\bibitem{W}  % Wajnryb 
B.~Wajnryb, 
{\it A simple relation for the mapping class group of an orientable surface}, 
Israel J. Math. {\bf 45} (1983), 157--174.  
\end{thebibliography} 
\end{document}