反平行双辫纽结微分展开式的因式分解
Factorization of differential expansion for antiparallel double-braid knots
摘要：
继续寻求独家拉卡矩阵，这是需要在陈-西蒙斯理论的有色树状结多项式的评价，我们建议提取它们从一种新的双进化-的反平行双辫子，这是一个简单的两个参数家庭的两桥结，推广的单参数家庭的扭曲结。在矩形表示$R=[r^s]$的情况下，我们发现了一个证据，即相应的微分展开奇迹般地因式分解，并可以从扭结得到。这减少了问题的矩形排他拉卡构造的答案只是一些扭曲的结。我们开发了一个最近的猜想，对这个家庭的最简单的成员（三叶草和8字结）的微分膨胀的结构，并提供了详尽的答案为第一个未知的情况下，$R=[33]$。答案包括任意扭曲和双辫结的HOMFLY和Racah矩阵$\bar S$和$S$ --允许计算任意树状（双发）结的$[33]$ -着色多项式。对于通用的矩形表示，详细描述的只是单层和双层金字塔的贡献，进行的方式是明确说明的例子$R=[44]$和$R=[55]$。这解决了问题的困难部分，但最后一个繁琐的步骤显式公式通用排他矩形拉卡矩阵仍然有待作出。
英文摘要：
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations $R=[r^s]$ we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of $R=[33]$. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices $\bar S$ and $S$ -- what allows to calculate $[33]$-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations described in detail are only the contributions of the single- and two-floor pyramids, the way to proceed is explicitly illustrated by the examples of $R=[44]$ and $R=[55]$. This solves the difficult part of the problem, but the last tedious step towards explicit formulas for generic exclusive rectangular Racah matrices still remains to be made.
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