diff --git a/sections/representations.tex b/sections/representations.tex
@@ -37,10 +37,14 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g
denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v =
\lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \mu_1, \mu_2,
\gamma, \eta_1, \ldots, \eta_{b - 1} \subset S_2^b\) be the curves of the
- Lickorish generators from Theorem~\ref{thm:lickorish-gens}.
- \begin{center}
- \includegraphics[width=.25\linewidth]{images/lickorish-gens-gen-2.eps}
- \end{center}
+ Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in
+ Figure~\ref{fig:lickorish-gens-genus-2}.
+ \begin{figure}
+ \centering
+ \includegraphics[width=.2\linewidth]{images/lickorish-gens-gen-2.eps}
+ \caption{The Lickorish generators for $g = 2$.}
+ \label{fig:lickorish-gens-genus-2}
+ \end{figure}
If \(n = 1\) then \(\rho(\Mod(S_2^b)) \subset \GL_1(\mathbb{C}) =
\mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by
@@ -243,10 +247,14 @@ representations.
Theorem~\ref{thm:lickorish-gens}. Once again, let \(L_\alpha =
\rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of
\(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(R \cong S_{g -
- 1}^1\) be the closed subsurface highlighted in the following picture.
- \begin{center}
- \includegraphics[width=.5\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
- \end{center}
+ 1}^1\) be the closed subsurface highlighted in
+ Figure~\ref{fig:korkmaz-proof-subsurface}.
+ \begin{figure}[ht]
+ \centering
+ \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
+ \caption{The subsurface $R \subset S_g^b$.}
+ \label{fig:korkmaz-proof-subsurface}
+ \end{figure}
% TODO: Add more comments on the injectivity of this map?
We claim that it suffices to find a \(m\)-dimensional