  This is an investigation into a classification of embeddings of a surface in
Euclidean $3$-space. Specifically, we consider $\mathbb{R}^3$ as having the
product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $\pi:\mathbb{R}^2
\times \mathbb{R} \to \mathbb{R}^2$ be the natural projection map onto the
Euclidean plane. Let $ \varepsilon : S_g \hookrightarrow \mathbb{R}^2 \times
\mathbb{R}$ be a smooth embedding of a closed oriented genus $g$ surface such
that the set of critical points for the map $\pi \circ \varepsilon$ is a smooth
(possibly multi-component) $1$-manifold, $\mathscr{C} \subset S_g$. We say
$\mathscr{C}$ is the crease set of $\varepsilon$ and two embeddings are in the
same isotopy class if there exists an isotopy between them that has
$\mathscr{C}$ being an invariant set. The case where $\pi \circ
\varepsilon|_\mathscr{C}$ restricts to an immersion is readily accessible,
since the turning number function of a smooth curve in $\mathbb{R}^2$ supplies
us with a natural map of components of $\mathscr{C}$ into $\mathbb{Z}$. The
Gauss-Bonnet Theorem beautifully governs the behavior of $\pi \circ \varepsilon
(\mathscr{C})$, as it implies $\chi(S_g) = 2 \sum_{\gamma \in \mathscr{C}}
t(\pi \circ \varepsilon (\gamma))$, where $t$ is the turning number function.
Focusing on when $S_g \cong S^2$, we give a necessary and sufficient condition
for when a disjoint collection of curves $\mathscr{C} \subset S^2$ can be
realized as the crease set of an embedding $\varepsilon: S^2 \hookrightarrow
\mathbb{R}^2 \times \mathbb{R}$. From there, we give the classification of all
isotopy classes of embeddings when $\mathscr{C} \subset S^2$ and
$|\mathscr{C}|=3$ -- a simple yet enlightening case. As a teaser of future
work, we give an application to knot projections and discuss directions for
further investigation.
