Posted on Categories:Topology, 拓扑学, 数学代写

# 数学代写|拓扑学代写TOPOLOGY代考|AMAT540

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|拓扑学代写TOPOLOGY代考|Tight and overtwisted

In this section we are going to discuss a fundamental dichotomy of contact structures on 3-manifolds, introduced by Eliashberg [64], namely, the division of contact structures into tight and overtwisted ones. At first sight, the definition of theses types of contact structures looks slightly peculiar. We shall see later what motivated this definition, and why it has proved seminal for the development of 3 -dimensional contact topology.

Recall the definition of the standard overtwisted contact structure $\xi_{\text {ot }}$ on $\mathbb{R}^3$ given in Example 2.1.6. This was described by the equation (in cylindrical coordinates)
$$\cos r d z+r \sin r d \varphi=0$$
Let $\Delta$ be the $\operatorname{disc}{z=0, r \leq \pi} \subset \mathbb{R}^3$. The boundary $\partial \Delta$ of this disc is a Legendrian curve for $\xi_{\text {ot }}$, in fact, the disc $\Delta$ is tangent to $\xi_{\text {ot }}$ along its boundary. This means that the characteristic foliation $\Delta_{\xi_{\mathrm{ot}}}$ consists of all radial lines, with singular points at the origin and at all boundary points (Figure 4.9).

If the interior of $\Delta$ is pushed up slightly, the singular points at the boundary can be made to disappear. Only the singular point at the centre remains, and the characteristic foliation now looks as in Figure 4.10, with $\partial \Delta$ a closed leaf of the characteristic foliation $\Delta_{\xi}$. (We shall prove this presently by an explicit calculation in a related case.) We call $\Delta$ (in its perturbed or unperturbed form) the standard overtwisted disc.

For our discussion in the following chapter, it is useful to describe the properties of $\Delta$ in terms of the contact framing and the surface framing of Legendrian knots (Defns. 3.5.1 and 3.5.2).

## 数学代写|拓扑学代写TOPOLOGY代考|Surfaces in contact $3-$ manifolds

We now want to take a more systematic look at surfaces in contact $3-$ manifolds, with a view towards using them as a tool in the classification of contact structures. Obviously some of the material on hypersurfaces in contact manifolds of arbitrary dimension (Section 2.5.4) will be relevant here. I am going to reiterate some of the arguments from that section in the special 3-dimensional setting to spare the reader from having to leaf back and forth. Throughout I assume that $M$ is a 3 -manifold with oriented and cooriented contact structure $\xi=\operatorname{ker} \alpha$ (with $d \alpha$ defining the orientation of $\xi$ ), and $S \subset M$ an oriented surface embedded in $M$. Occasionally we allow $S$ to have boundary, but then there will be some control over the boundary, e.g. if it consists of Legendrian curves. All results can typically be proved for non-orientable surfaces by passing to a double cover.

As in Section 2.5.4 we identify a neighbourhood of $S$ in $M$ with $S \times \mathbb{R}$, and $S$ with $S \times{0}$, where we write $z \nmid$ for the $\mathbb{R}$-coordinate. We make this identification compatible with orientations: the orientation of $S$ followed by the natural orientation of $\mathbb{R}$ gives the orientation of $M$ (induced by $\xi$ ). We write the contact form $\alpha$ as
$$\alpha=\beta_z+u_z d z$$
where $\beta_z, z \in \mathbb{R}$, is a smooth family of 1 -forms on $S$, and $u_z: S \rightarrow \mathbb{R}$, a smooth family of functions. Then
$$d \alpha=d \beta_z-\dot{\beta}_z \wedge d z+d u_z \wedge d z$$
where the dot denotes the derivative with respect to $z$. Thus, the contact condition becomes
$$u_z d \beta_z+\beta_z \wedge\left(d u_z-\dot{\beta}_z\right)>0,$$
meaning that the $2-$ form on the left is a positive area form on $S$.

## 数学代写|拓扑学代写TOPOLOGY代考|Tight and overtwisted

$$\cos r d z+r \sin r d \varphi=0$$

## 数学代写|拓扑学代写TOPOLOGY代考|Surfaces in contact $3-$ manifolds

$$\alpha=\beta_z+u_z d z$$

$$d \alpha=d \beta_z-\dot{\beta}_z \wedge d z+d u_z \wedge d z$$

$$u_z d \beta_z+\beta_z \wedge\left(d u_z-\dot{\beta}_z\right)>0,$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。