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