memoire-m2

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -87,11 +87,10 @@ bitorus \(S_2\).
 \begin{definition}
   Given a simple closed curve \(\alpha \subset S\), fix a closed annular
   neighborhood \(A \subset S\) of \(\alpha\) -- i.e. \(A \cong \mathbb{S}^1
-  \times [0, 1]\). Let \(f \in \Mod(A) \cong \Mod(\mathbb{S}^1 \times [0, 1])\)
-  be as in Example~\ref{ex:mcg-annulus}. The \emph{Dehn twist \(\tau_\alpha \in
-  \Mod(S)\) about \(\alpha\)} is defined as the image of the generator \(f \in
-  \Mod(A) \cong \mathbb{Z}\) under the inclusion homomorphism \(\Mod(A) \to
-  \Mod(S)\).
+  \times [0, 1]\). Let \(f \in \Mod(A) \cong \Mod(\mathbb{S}^1 \times [0, 1])
+  \cong \mathbb{Z}\) be the generator from Example~\ref{ex:mcg-annulus}. The
+  \emph{Dehn twist \(\tau_\alpha \in \Mod(S)\) about \(\alpha\)} is defined as
+  the image of \(f\) under the inclusion homomorphism \(\Mod(A) \to \Mod(S)\).
 \end{definition}
 
 \begin{figure}[ht]
@@ -112,11 +111,10 @@ classes that permute the punctures of \(S\).
   Given an arc \(\alpha \subset S\) joining two punctures in the interior of
   \(S\), fix a closed neighborhood \(D \subset S\) of \(\alpha\) with \(D \cong
   \mathbb{D}^2 \setminus \{-\sfrac{1}{2}, \sfrac{1}{2}\}\). Let \(f \in
-  \Mod(\mathbb{S}^1 \times [0, 1]) \cong \Mod(D)\) be as in
-  Example~\ref{ex:mcg-twice-punctured-disk}. The \emph{half-twist \(h_\alpha
-  \in \Mod(S)\) about \(\alpha\)} is defined as the image of the generator \(f
-  \in \Mod(D) \cong \mathbb{Z}\) under the inclusion homomorphism \(\Mod(D) \to
-  \Mod(S)\).
+  \Mod(\mathbb{S}^1 \times [0, 1]) \cong \Mod(D) \cong \mathbb{Z}\) be the
+  generator from Example~\ref{ex:mcg-twice-punctured-disk}. The
+  \emph{half-twist \(h_\alpha \in \Mod(S)\) about \(\alpha\)} is defined as the
+  image of \(f\) under the inclusion homomorphism \(\Mod(D) \to \Mod(S)\).
 \end{definition}
 
 It is interesting to study how the geometry of two curves affects the