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 $\mathcal{Z}$--$F_{Tw}$ decomposition with the twist-knot $F$-factors and non-standard $\mathcal{Z}$-factors and a discovery of still another triangular and universal transformation $V$, which converts $\mathcal{Z}$ to the standard $Z$-factors $V^{-1}\cdot \mathcal{Z}=Z$ and allows to calculate $F$ as $F=V\cdot F_{Tw}$.

14 pages