Posted on Categories:Knot Theory, 扭结理论, 数学代写

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## 数学代写|扭结理论代写Knot Theory代考|Computing arc lengths and areas

2.1.2. Computing arc lengths and areas. Now we will use the formulas obtained above to do calculations, in order to better understand the hyperbolic space $\mathbb{H}^2$.

Example 2.1. Fix a height $h>0$, and consider first a horizontal line segment between points $(0, h)=i h$ and $(1, h)=1+i h$ in $\mathbb{H}^2$. We may parameterize the line segment by $\gamma(t)=(t, h)$, for $t \in[0,1]$. Using equation (2.1), we find that the arc length of $\gamma$ is $|\gamma|=1 / h$. Note that because $h$ is fixed, the arc length of $\gamma$ is just its usual Euclidean length rescaled by $1 / h$. Thus when $h=1$, the length of $\gamma$ is 1 . When $h$ becomes large, the arc length becomes very small. In other words, points with the same height become very close together as their heights increase. On the other hand, as $h$ approaches 0 , the length of $\gamma$ approaches infinity. In fact, points near the real line $\mathbb{R}=\left{(x, 0) \in \mathbb{R}^2\right}$ can be very far apart.

Example 2.2. Now consider a vertical line between points $(x, a)$ and $(x, b)$, for $x, a, b$ fixed in $\mathbb{R}, 0<a<b$. Such a line can be parameterized by $\zeta(t)=(x, t)$ for $t \in[a, b]$. So $\zeta^{\prime}(t)=(0,1)$. Thus its arc length is given by
$$|\zeta|=\int_a^b \sqrt{0+1} \frac{1}{s} d s=\log \left(\frac{b}{a}\right) .$$
If we set $b=1$ and let $a$ approach 0 , note that the arc length of $\zeta$ gets arbitrarily large, approaching infinity. Similarly, setting $a=1$ and letting $b$ approach infinity give arbitrarily long lengths.

The real line $\mathbb{R}=\left{(x, 0) \in \mathbb{R}^2\right}$ along with the point at infinity $\infty$ play an important role in the geometry of $\mathbb{H}^2$, although these points are not contained in $\mathbb{H}^2$.

## 数学代写|扭结理论代写Knot Theory代考|Geodesics and isometries

Geodesics and isometries. Recall that a geodesic between points $p$ and $q$ is a length minimizing curve between those points. An infinite geodesic is a curve $\gamma$ from $\mathbb{R}$ to a Riemannian manifold such that for any $s<t \in \mathbb{R}$, the curve $\gamma([s, t])$ minimizes the distance between $\gamma(s)$ and $\gamma(t)$.
Theorem 2.5. The infinite geodesics in $\mathbb{H}^2$ consist of vertical straight lines and semi-circles with center on the real line.

Note these are exactly the circles and lines in the upper half-plane that meet $S_{\infty}^1$ at right angles. See Figure 2.2. Observe that between any two points in the upper half-plane, there is a unique vertical line or semi-circle between them. Thus a geodesic between points $p$ and $q$ in $\mathbb{H}^2$ is a segment of a semi-circle or a vertical straight line. An infinite geodesic can also be viewed as the unique semi-circle or vertical straight line between two points on the boundary at infinity of $\mathbb{H}^2$. We will typically drop the word “infinite” to describe geodesics between points on the boundary at infinity. Thus we use the same word “geodesic” to describe both infinite or bounded arcs, depending on context.

The proof of Theorem $2.5$ is left as an exercise in Riemannian geometry. The simplest way to prove the theorem uses coordinates and a bit more Riemannian geometry than we have reviewed so far. The interested reader can work through the details. The fact that these are the geodesics of $\mathbb{H}^2$ is all we will need going forward.

## 数学代写|扭结理论代写Knot Theory代考|Computing arc lengths and areas

2.1.2. 计算弧长和面积。现在我们将使用上面得到的公式进行计算，以便更好地理解双曲空间 $\mathbb{H}^2$. $\gamma(t)=(t, h)$ ，为了 $t \in[0,1]$. 使用等式 (2.1)，我们发现弧伥 $\gamma$ 是 $|\gamma|=1 / h$. 请注意，因为 $h$ 是固定的，弧长为 $\gamma$ 只是它通常的 欧几里得长度重新缩放 $1 / h$. 因此当 $h=1$ ，的长度 $\gamma$ 是 1 。什么时候 $h$ 变大，弧长变得非常小。换句话脱，具有相同高度的点随着 高度的垾加变得非常接近。另一方面，如 $h$ 接近 0 ，长度为 $\gamma$ 接近无穷大。事实上，靠近实线的点 lleft 的分隔符缺失或无法识别 可以相距很远。

$$|\zeta|=\int_a^b \sqrt{0+1} \frac{1}{s} d s=\log \left(\frac{b}{a}\right) .$$

## 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。

Posted on Categories:Knot Theory, 扭结理论, 数学代写

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## 数学代写|扭结理论代写Knot Theory代考|Shrink the knot to ideal vertices for the bottom polyhedron

Notice that underneath the knot, the picture of faces, edges, and vertices will be slightly different. In particular, when finding the top polyhedron, we collapsed overstrands to a single ideal vertex. When you put your head underneath the knot, what appear as overstrands from below will appear as understrands on the usual knot diagram.

One way to see this difference is to take the 3-dimensional model constructed in Figure 1.4. Figure 1.3 shows the view of the faces meeting at an edge from the top. If you turn the model over to the opposite side, you will see how the faces meet underneath. Figure $1.9$ illustrates this. Note that $U$ now meets $V$, and $S$ meets $T$.

In terms of the combinatorics, edges of Figure $1.5$ that are isotopic by sliding an endpoint along an understrand are identified to each other on the bottom polyhedron, but edges only isotopic by sliding an endpoint along an overstrand are not identified.

As above, collapse each knot strand corresponding to an understrand to a single ideal vertex. The result is Figure 1.10.

One thing to notice: we sketched the top polyhedron with our heads inside the ball on top, looking out. If we move the face $D$ away from the point at infinity, then it wraps above the other faces shown in Figure 1.8.
On the other hand, we sketched the bottom polyhedron with our heads outside the ball on the bottom. If we move the face $D$ away from the point at infinity, it wraps below the other faces shown in Figure 1.10.

## 数学代写|扭结理论代写Knot Theory代考|Rebuilding the knot complement from the polyhedra

1.1.5. Rebuilding the knot complement from the polyhedra. Figures $1.8$ and $1.10$ show two ideal polyhedra that we obtained by studying the figure- 8 knot complement. We claim that they glue to give the figure- 8 knot complement. That is, attach face $A$ on the bottom polyhedron to the face labeled $A$ on the top polyhedron, ensuring that the edges bordering face $A$ match up. Similarly for the other faces.

This process of gluing faces and edges gives exactly the complement of the knot. By construction, faces glue to give the faces illustrated in Figure 1.6, and edges glue to give the edges there, except that when we have finished, all four edges in an isotopy shown in that figure have been glued together.

## 数学代写|扭结理论代写Knot Theory代考|Rebuilding the knot complement from the polyhedra

1.1.5。从多面体重建结补。数字1.8和1.10显示我们通过研究图 8 结补获得的两个理想多面体。我们声称它们粘合以提供数字 8 结补码。也就是贴脸一个在底部多面体上标记的面一个在顶部多面体上，确保边缘与面接壤一个配对。其他面孔也是如此。

## 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。

Posted on Categories:Knot Theory, 扭结理论, 数学代写

## 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！

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## 数学代写|扭结理论代写Knot Theory代考|A Brief Introduction to Hyperbolic Knots

This book gives an introduction to knots, links, and hyperbolic geometry. Before we begin, we need to carefully define what we mean by knots and links, and that is done in this chapter. We also introduce classical problems in knot theory, and problems motivated by geometry, especially hyperbolic geometry. This chapter is meant to motivate future chapters, and it has many references to content covered in more detail later in the book, where we address some of these problems. Many of the questions described in this chapter have partial answers, and many are still wide open.

## 数学代写|扭结理论代写Knot Theory代考|An introduction to knot theory

The earliest study of knots seems to be by Gauss, Listing, and especially Tait, who published several papers on knot theory in the years 1876 through 1885. In a preface to his work on knot theory, republished in his 1898 Scientific papers [Tai98], Tait writes:
“The subject [knot theory] is a very much more difficult and intricate one than at first sight one is inclined to think, and I feel that I have not succeeded in catching the keynote.”
Since Tait’s work, advances in knot theory have come through applications of topology, algebra, and invariants arising in quantum field theory, but no single mathematical field has led to simple tools that apply to all knots. In other words, perhaps mathematicians still have not succeeded in catching the “key-note.” Perhaps there is no “key-note” in knot theory.

However, there are definitely mathematical techniques that work well when applied to particular problems or particular families. This book introduces techniques arising from geometry.

## 数学代写|扭结理论代写Knot Theory代考|An introduction to knot theory

“这个主题 [结理论] 比乍一看的人倾向于认为的要困难和复杂得多，我觉得我没有成功抓住主题。”

## 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。