论文标题
$ \ mathbb {r}^2 \ times \ mathbb {r} $中的表面嵌入
Surface embeddings in $\mathbb{R}^2\times\mathbb{R}$
论文作者
论文摘要
这是对欧几里得表面嵌入的分类的调查，$ 3 $ - 空间。 Specifically, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $π:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2$ be the natural projection map onto the Euclidean plane.令$ \ varepsilon：s_g \ hookrightArrow \ mathbb {r}^2 \ times \ times \ mathbb {r} $是对封闭的定向属$ g $表面的平稳嵌入，使得$ $π\ circ \ varepsilon $的关键点集合是一个流畅的（可能是$ 1 $ 1 $ 1 $ 1 $ - \子集S_G $。我们说$ \ mathscr {c} $是$ \ varepsilon $的折痕集，如果它们之间存在$ \ mathscr {c} $是不变的集合，则两个嵌入在同一同位素类中。 $π\ circ \ varepsilon | _ \ mathscr {c} $限制了沉浸式的情况，因为在$ \ mathbb {r}^2 $中平滑曲线的转弯数函数可为我们提供$ \ \ nathscr $ \ nathscr {c} $的自然图。 Gauss-bonnet定理完美地控制了$π\ Circ \ Varepsilon（\ Mathscr {C}）$的行为，因为它暗示了$χ（s_g）= 2 \ sum_ {γ{γ\ in \ Mathscr {c}}}} t（c}} t（π\ circ \ circ \ circ \ varepsilon（prict）$ there（prict）专注于$ s_g \ cong s^2 $，我们给出了必要和足够的条件，以何时偏离曲线$ \ mathscr {c} \ subset s^2 $，可以实现为嵌入的$ \ \ varepsilon：s^2 \ hookrightArrow \ hookrow \ mathbb \ mathbb {r} $ {r} $ nations $ \ varepsilon：s^2 \ mathbb {$从那里开始，当$ \ mathscr {c} \ subset s^2 $和$ | \ mathscr {c} | = 3 $时，我们给出了所有同位素类嵌入的分类。作为未来工作的预告片，我们提出了打结预测的申请，并讨论了进一步调查的方向。
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 $π:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2$ 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 $π\circ \varepsilon$ 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 $π\circ \varepsilon|_\mathscr{C}$ 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 $π\circ \varepsilon (\mathscr{C})$, as it implies $χ(S_g) = 2 \sum_{γ\in \mathscr{C}} t(π\circ \varepsilon (γ))$, 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.
