辫群 电影

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A journey through the mathematical theory of braids.

The movie is divided into four chapters, each of about 15 minutes.

The first contains the basic concepts: the formalization of braids, the group structure on the set of braids and Artin presentation of the braid group. A braid is now described by a word on a set of letters, the generators.

The second chapter deals with the word problem: when do two words represent the same braid? Two algorithms are presented to solve this problem, the Artin braid combing and the handle reduction.

In the third chapter knots are presented and put in relation with braids. In the final part the Jones polynomial is introduced: it is a powerful knot invariant that was defined through braids.

The last chapter describes braids as dances, that is, motions of points in the disc. The Hilden group, a subgroup of the braid group, is defined and related to another way of closing braids to obtain knots.

纽结理论之旅。 影片分为四章，每章约15分钟。
第一章包含基本概念：辫群的群结构和纽结群的 Artin 表示。生成子。
第二章处理词的问题：两个词什么时候代表同一条辫子？提出了两种算法来解决这个问题，Artin 合并和手柄约减。
在第三章中，介绍了纽结，它与辫子有关。最后一部分介绍了琼斯多项式：它是一个通过纽结定义的强大的结不变量。
最后一章将辫子描述为舞蹈，即圆盘中点的运动。 Hilden 群是辫群的一个子群，它被定义并与另一种闭合辫子以获得纽结的方式相关。

The braid group on $n$ strands admits a presentation with generators $\sigma_1,\sigma_2,⋯,\sigma_{n-1}$ and relations
$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $1≤i≤n-2$;
$\sigma_i\sigma_j=\sigma_j\sigma_i$ for $|i-j|>1$.

Czhang271828

bug 频频, 改日再研究
图中 $c$ 为交换约束

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回复 2# Czhang271828
使用\xymatrix{...}
比如这个手册的第2页的例子:
$\xymatrix{
A &*+[F]{\sum_{i=n}^m {i^2}} \\
& {\bullet} & D \ar[ul] }$
使用\begin{xy}...\end{xy}
比如这个幻灯片的第21页
\begin{xy}
(0,0)*+{A}="A";
(20,0)*+{B}="B";
**\crv{(5,10)&(15,-10)};
\end{xy}
\begin{xy}
(-20,0)*+{\cdots},{\ar (-16,0);(-6,0)};
(0,-14.2)*+{H^{i+1}(A)}="target",
(59,-14.2)*+{\cdots},{\ar (48,-14.2);(55,-14.2)};
{\ar (7,-14.2);(13,-14.2)};
(20,-14.2)*+{H^{i+1}(B)},{\ar (27,-14.2);(33,-14.2)};
(40,-14.2)*+{H^{i+1}(C)},(0,0)*+{H^i(A)};
(20,0)*+{H^i(B)};{\ar (6,0);(14,0)};
(40,0)*+{H^i(C)}="C";{\ar (26,0);(34,0)};
\PATH ~={**\dir{-}?>*\dir{}}~>{|>*\dir{>}}
\end{xy}
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Knots and Primes

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f-lohmueller.de/pov_tut/animate/anim200d.htm
POV-RAY 动画