论文标题
差分扩展的观点
Perspectives of differential expansion
论文作者
论文摘要
我们概述了彩色结多项式的差分扩展（DE）的当前状态，即其$ z $ - $ f $ $ f $分解为表示形式以及依赖性零件。它的存在是在对称和反对称表示中的homfly-pt多项式定理，但是超越的一切仍然是假设的，而且很难探索和解释。但是，DE仍然是现代结理论中知识和计算手段的主要来源之一。我们专注于以下主题：DE对非平凡结的适用性，其修改对具有非变化缺陷的结的修改，而对于非矩形表示。一个基本的新颖性是对具有twist-knot $ f $ -f $ - f $ - f $ - f $ - f $ - f $ f $ f $ - $ {\ cal z} $的分析的分析 - $ {f _ {f_ {tw}} $分解 - $ {\ cal z} $ - 因素，并且发现了另一个三角形和通用变换式$ v $ $ v $，$ v $ $ \ z} $ v^{ - 1} \ cdot {\ cal z} = z $，并允许将$ f $计算为$ f = v \ cdot f_ {tw} $。
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 $F = V\cdot F_{Tw}$.
