diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -8,9 +8,7 @@ intuition behind them, culminating in the statement of a presentation for
geometry of curves in \(\Sigma_g\) -- see
Theorem~\ref{thm:wajnryb-presentation}.
-We start by the so called \emph{lantern relation}.
-
-\begin{fundamental-observation}
+\begin{fundamental-observation}[Lantern relation]
Let \(\Sigma_0^4\) be the surface the 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
@@ -320,7 +318,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden
\cite[Section~9.4]{farb-margalit}.
\end{observation}
-\begin{fundamental-observation}
+\begin{fundamental-observation}[$k$-chain relations]
The Birman-Hilden isomorphism \(\SMod(\Sigma_\ell^1) \isoto \Mod(\Sigma_{0,
2\ell+1}^1)\) takes the twists \(\tau_\delta \in \SMod(\Sigma_\ell^1)\) about
the boundary \(\delta = \partial \Sigma_\ell^1\) to \(\tau_{\bar\delta}^2 \in
@@ -368,7 +366,7 @@ to obtain other relations. Since \(\iota\) has \(2g + 2\) fixed points in
2})\).
\end{theorem}
-\begin{fundamental-observation}
+\begin{fundamental-observation}[Hyperelliptic relations]
Let \(\alpha_1, \ldots, \alpha_{2g}, \delta \subset \Sigma_g\) be as in
Figure~\ref{fig:hyperellipitic-relations}. Then
\begin{equation}\label{eq:hyperelliptic-eq}
diff --git a/sections/twists.tex b/sections/twists.tex
@@ -161,7 +161,7 @@ too.
f^{-1}\).
\end{observation}
-\begin{observation}
+\begin{observation}[Disjointness relations]
Given \(f \in \Mod(\Sigma)\), \([f, \tau_\alpha] = 1 \iff f \cdot [\alpha] =
[\alpha]\). In particular, \([\tau_\alpha, \tau_\beta] = 1\) for \(\alpha\)
and \(\beta\) disjoint.
@@ -174,7 +174,7 @@ too.
[\alpha] = [\beta]\) and then apply Observation~\ref{ex:conjugate-twists}.
\end{observation}
-\begin{fundamental-observation}\label{ex:braid-relation}
+\begin{fundamental-observation}[Braid relations]\label{ex:braid-relation}
Given \(\alpha, \beta \subset \Sigma\) with \(\#(\alpha \cap \beta) = 1\), it
is not hard to check that \(\tau_\beta \tau_\alpha \cdot [\beta] =
[\alpha]\). From Observation~\ref{ex:conjugate-twists} we then get