Posted on Categories:Low Dimensional Topology, 低维拓扑, 数学代写

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## 数学代写|低维拓扑代写Low Dimensional Topology代考|The 2-fold Branched Cover of S4 Branched Along a Knotted or Linked Surface

In [2] a Seifert algorithm for knotted surfaces that project without branch points is presented. This algorithm was adjusted by the second author in [9] to be applied to the surface braid case. In our respective books, [3] or $[10]$ these algorithms are described in detail. The 2-fold branched cover of $S^4$ branched along an orientable knotted or linked surface can be constructed from two copies of $S^4$ both cut along the Seifert solid, and gluing the positive side of one copy to the negative side of the other.

The braid chart can be used to give a movie description of the branch locus. In such a movie, or in a standard movie description of the knotted surface, the Seifert solid is cut into 2-dimensional slices that are Seifert surfaces for the classical cross-sectional links. Thus for any such cross-section, the previous section provided an embedding of the 2-fold branched cover of that cross-section. These are, then, connected by either handle attachments or isotopies that are induced by the Reidemeister moves.

Specifically, the birth of a simple closed curve corresponds to a 1-handle attached between the successive 3-dimensional 2-fold branched covers. A 1-handle attached between successive movie stills corresponds to a 2-handle attached between successive 3-dimensional 2-fold branched covers.

Following the discussion of the 3 -fold branched covers of classical knots, an example of 3 -fold branched cover of the 4-sphere branched along the spun trefoil will be given. These examples are more complicated than the 2-fold branched coverings.

In Fig. 25 we indicate that the 2-fold branched cover of $S^4$ branched along an unknotted sphere is $S^4$, and we indicate an embedded folding. Meanwhile, the 2-fold branched cover of $S^4$ branched along an unknotted torus is $S^2 \times S^2$ and also has an embedded folding.

## 数学代写|低维拓扑代写Low Dimensional Topology代考|Three-fold Simple Branched Covers

In this section, we construct embeddings and immersions of 3 -fold branch covers of the 3- and 4-sphere branched over the trefoil, the knot $7_4$, and the knotted sphere that is the spin of the trefoil. We begin with the trefoil as indicated Fig. 26. The trefoil is 3-colorable, and we can lift this coloring to a representation to the 3-string braid group (also the fundamental group of the knot itself) by assigning the braid generator $\sigma_1$ to the (blue) label 1, assigning the braid generator $\sigma_2$ to the (red) label 2, and assigning $\sigma_2^{-1} \sigma_1 \sigma_2$ to the (green) label 3. From this coloring, a sequence of braid charts is constructed. The endpoints of the arcs in the charts correspond to black vertices, and as the charts are stacked, these end points trace the outline of the knot. Successive braid charts differ by a planar isotopy, by one of the chart moves, or by the addition or subtraction of a simple arc. As before, the addition or subtraction of an arc corresponds to attaching a handle between successive sheets in the covering. The sequence of charts can be interpolated in 3-space to create a (non-generically) immersed surface in 3-space that we call a curtain. To construct the 3 -fold branch cover, we take three copies of $S^3$, cut each along the appropriately labeled sheets of the curtains, and re-glue. The embedding in this case is achieved by interpreting each chart as a braided surface and interpolating between successive surfaces.

## 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:Low Dimensional Topology, 低维拓扑, 数学代写

## avatest™帮您通过考试

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## 数学代写|低维拓扑代写Low Dimensional Topology代考|Lifting a permutation movie to a braid movie

A braid chart of degree $n$ is a permutation chart in which orientations on the edges have been chosen so that white vertices and crossings are of the form indicated in the figure. Thus there are three incoming and three outgoing edges that alternate $i, i \pm 1, i, i \pm 1, i, i \pm 1$ in cyclic order. Similarly, at a crossing the edges with labels $i$ and the edges with labels $j$ are oriented consistently $(|i-j|>1)$ as indicate in Fig. 14. In this case, we say that there is a flow though the white vertex or the crossing. If a permutation chart can be consistently oriented to create a braid chart, then the permutation movie can be lifted to a braid movie. Specifically, right pointing edges with label $i$ correspond to braid generators $\sigma_i$, and left pointing edges with label $i$ correspond to $\sigma_i^{-1}$.

Let $\Gamma_n$ denote a degree $n$ permutation chart. If there is a consistent orientation on the edges of $\Gamma_n$, then the opposite orientation will also be consistent; we let $\Gamma_n^{\pm}$denote the resulting braid charts. As before, let $M^2(\Gamma)$ denote the $n$-fold irregular simple branched cover of $D^2$ that is associated to the chart $\Gamma=\Gamma_n$, and let $\hat{M}(\Gamma)$ denote its extension to the 2 -sphere.

Theorem 3.5. If $\Gamma_n^{\pm}$is a braid chart, then there is an embedding, $\tilde{F}: \hat{M}(\Gamma) \rightarrow S^2 \times[0,1] \times[0,1]$ of the $n$-fold irregular simple branched cover $F: \hat{M}(\Gamma) \rightarrow S^2$ such that the composition $p \circ \tilde{F}$ agrees with the covering map $F$ where $p: S^2 \times[0,1] \times[0,1] \rightarrow S^2$ is the projection onto the first factor. In this way the covering $F$ has a folded embedding.

Proof. The lifting of each permutation to a braid induces an embedding of the surface $\hat{M}$ into $S^2 \times[0,1] \times[0,1]$. The projection to the first two factors $S^2 \times[0,1]$ induces the generic map of Proposition 3.4.

## 数学代写|低维拓扑代写Low Dimensional Topology代考|A partial lifting

It is possible that a given chart cannot be oriented consistently so that all white vertices have a flow. A semi-oriented chart is a permutation chart $\Gamma^*$ that includes a fourth type of vertex which is bivalent, and the chart is oriented such that (1) each bivalent vertex is either a source or a sink and (2) each white vertex and each crossing has a flow. At a source vertex the two emanating edges have the same label and point away from the vertex. At a $\operatorname{sink}$ the edges point towards the vertex.
Lemma 3.6. Any permutation chart can be semi-oriented.
Proof. Locally orient the edges so that there is a flow through all white vertices and all crossings. If there is an edge whose endpoints are either white vertices or crossings such that the local orientations do not match, then introduce a source or sink. Do the same for all such edges.

## 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:Low Dimensional Topology, 低维拓扑, 数学代写

## avatest™帮您通过考试

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## 数学代写|低维拓扑代写Low Dimensional Topology代考|Moves that Change the Topology of the Underlying Foam

It is important to remark that not only are embedded trivalent graphs studied for their own sake, but a given knotted handlebody in 3-space deformation retracts to an embedded trivalent graph. The graph, however, is not unique. Two graphs that “carry” such a knotted handlebody differ by the so-called IH-move. Up to equivalence, the $\mathrm{IH}$-move is given via the movie parametrization of the basic foam $Y^2$. The theory of knotted handlebodies embedded in 3 -space is equivalent to the theory of knotted trivalent graphs modulo the IH-move.

A similar situation holds in 4-space. We can include among the Roseman moves two additional moves that are indicated below. The first of these is the invertibility of the IH-move. In the theory of special spines for 3-manifolds it is sometimes called the lune move or the orthogonality condition. The reason for the latter name comes from the Tureav-Viro [15] invariants, the neighborhood of a vertex of a foam is colored by representations of $U_q\left(s l_2\right)$, and the move corresponds to the orthogonality condition for the $6 j$-symbol. See also [4]. Observe that if a foam is embedded in 3-space, then regular neighborhood of the foam is invariant under this move. So similarly, a regular neighborhood in 4-space of such a foam is also invariant since it can be obtained from the neighborhood in 3-space by the cartesian product with an open interval.

## 数学代写|低维拓扑代写Low Dimensional Topology代考|Future Work

This paper is a technical piece that is necessary for a serious study of knotted foams and their 3 (and higher)-dimensional generalizations. In work with Atsushi Ishii and Masahico Saito, we will establish a cohomology theory for certain algebraic systems that is sufficient to define nontrivial invariants of knotted 2-foams in 4-space.

The inclusion of the penultimate section is also meant to indicate the initial stages in the study of 3-dimensional foams. In particular, one can construct movies of 3-dimensional foams embedded in 5 -space by including the Roseman/Reidemeister moves of Theorem 1.1, the orthogonality and Eilliott-Beidenharn moves, critical points of surfaces and critical points of the edge sets. Thus the critical points of surfaces correspond to $0,1,2$, and 3-handles that are attached to the solid sheets of 3 -foams. Moreover, the edge set of a 3 -foam is a 2-foam. Important moves to 3 -foams are easy to establish. A full Roseman-type theorem is unknown to the author at this time tedious.

Finally, it is worth mentioning that there is an underlying categorical motivation here that is related to the tangle hypothesis of Baez and Dolan [2]. Here we are considering the interaction between a braiding and a Frobenius structure as well as the identities among relations of these. The precise location of knotted foams in the BaezDolan table is an interesting taxonomic problem.

## 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。