Abstract 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 Z – F T w decomposition with the twist-knot F-factors and non-standard Z -factors and a discovery of still another triangular and universal transformation V, which converts Z to the standard Z-factors V − 1 Z = Z and allows to calculate F as F = V F T w .

中文翻译：

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差异化扩张的观点

摘要 我们概述了彩色结多项式的微分展开 (DE) 的现状，即它们的 Z-F 分解为表示和结相关部分。它的存在是对称和反对称表示中 HOMFLY-PT 多项式的一个定理，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 仍然是现代结理论的主要知识来源和计算手段之一。我们专注于以下主题：DE 对非平凡结的适用性、其对具有非消失缺陷的结的修改以及用于非矩形表示的 DE。