微分膨胀的观点
Perspectives of differential expansion
摘要：
我们概述了目前的状态的差分展开（DE）的有色结多项式，即他们的$Z$ - $F$分解成代表性和结依赖的部分。它的存在性是HOMFLY-PT多项式在对称和反对称表示中的一个定理，但除此之外的一切仍然是假设的-并且很难探索和解释。然而，DE仍然是现代纽结理论的主要知识来源和计算方法之一。我们专注于以下主题：DE的适用性，非平凡的结，其修改结与非零缺陷和DE非矩形表示。一个基本的新颖性是分析了一个更简单的${\cal Z}$ - ${F_{Tw}}$分解与扭结$F$ -因素和非标准${\cal Z}$ -因素和发现了另一个三角和通用的转换$V$，它转换$\cal{Z}$到标准的$Z$ -因素$V^{-1}\cdot {\cal Z}= Z$，并允许计算$F$为$F = V\cdot F_{Tw}$。
英文摘要：
We outline the current status of the differential expansion (DE) of colored knot polynomials i.e. of their $Z$--$F$ decomposition into representation-- and knot--dependent parts. Its existence is a theorem for HOMFLY-PT polynomials in symmetric and antisymmetric representations, but everything beyond is still hypothetical -- and quite difficult to explore and interpret. However, DE remains one of the main sources of knowledge and calculational means in modern knot theory. We concentrate on the following subjects: applicability of DE to non-trivial knots, its modifications for knots with non-vanishing defects and DE for non-rectangular representations. An essential novelty is the analysis of a more-naive ${\cal Z}$--${F_{Tw}}$ decomposition with the twist-knot $F$-factors and non-standard ${\cal Z}$-factors and a discovery of still another triangular and universal transformation $V$, which converts $\cal{Z}$ to the standard $Z$-factors $V^{-1}\cdot {\cal Z}= Z$ and allows to calculate $F$ as $F = V\cdot F_{Tw}$.
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