diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -9,12 +9,12 @@ geometry of curves in \(\Sigma_g\) -- see
Theorem~\ref{thm:wajnryb-presentation}.
\begin{fundamental-observation}[Lantern relation]
- Let \(\Sigma_0^4\) be the surface the of genus \(0\) with \(4\) boundary
+ Let \(\Sigma_0^4\) be the surface of genus \(0\) with \(4\) boundary
components and \(\alpha, \beta, \gamma, \delta_1, \ldots, \delta_4 \subset
- \Sigma_0^4\) be as in Figure~\ref{fig:latern-relation}. Consider the
- surfaces \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and
- \(\Sigma_{0,1}^3 = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0
- \})\), as well as the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to
+ \Sigma_0^4\) be as in Figure~\ref{fig:latern-relation}. Consider the surfaces
+ \(\Sigma_0^3 = \Sigma_0^4 \cup_{\delta_1} \mathbb{D}^2\) and \(\Sigma_{0,1}^3
+ = \Sigma_0^4 \cup_{\delta_1} (\mathbb{D}^2 \setminus \{ 0 \})\), as well as
+ the map \(\operatorname{push} : \pi_1(\Sigma_0^3, 0) \to
\Mod(\Sigma_{0,1}^3)\). Let \(\eta_1, \eta_2, \eta_3 \subset \Sigma_0^3\) be
the loops from Figure~\ref{fig:lantern-relation-capped}, so that \([\eta_1]
\cdot [\eta_2] = [\eta_3]\). From Observation~\ref{ex:push-simple-loop} we