Perspectives of differential expansion

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 ${\cal Z}$--${F_{Tw}}$ 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 $F = V\cdot F_{Tw}$.