Perspectives of differential expansion

TL;DR

This paper reviews the current understanding of differential expansion in colored knot polynomials, exploring its applicability, modifications, and new transformations for complex knots and representations, highlighting recent theoretical advances.

Contribution

It introduces a novel transformation V that relates different decompositions of knot polynomials, expanding the theoretical framework of differential expansion beyond symmetric and antisymmetric cases.

Findings

Existence of a universal transformation V for decompositions

Extension of differential expansion applicability to complex knots

Introduction of a new ${ m Z}$--${ m F_{Tw}}$ decomposition

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 $F = V\cdot F_{Tw}$.