Perspectives of differential expansion

• Published in: arXiv.org
• Main Authors: Bishler, L, Morozov, A
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.06.2020
• Subjects: Decomposition; Existence theorems; Knot theory; Mathematical analysis; Mathematics - Geometric Topology; Mathematics - Mathematical Physics; Physics - High Energy Physics - Theory; Physics - Mathematical Physics; Polynomials; Representations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2006.01190

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 \(F = V\cdot F_{Tw}\).