Surface embeddings in $\mathbb{R}^2\times\mathbb{R}$

TL;DR

This paper classifies surface embeddings in  structured as , focusing on crease sets and their isotopy classes, with detailed results for spheres and applications to knot projections.

Contribution

It provides a necessary and sufficient condition for crease sets of embeddings of spheres in , and classifies embeddings with three crease curves, advancing understanding of surface embeddings.

Findings

Characterization of crease sets for sphere embeddings.

Classification of embeddings with three crease curves.

Application to knot projection analysis.

Abstract

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.