2020/06/01 by L. Bishler, Liudmila Bishler, А. Морозов +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Antisymmetric relation #Combinatorics #Connective tissue disorders research #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Mathematical physics #Mathematics #Physics #Pure mathematics #Twist #hep-th #math-ph #math.GT #math.MP
paper · pdf · doi:10.1016/j.physletb.2020.135639
published as Phys.Lett. B808 (2020) 135639 · 14 pages
arxiv created 2020/06/01 · openalex publication_date 2020/07/27 · arxiv updated 2021/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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–FTw 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−1Z=Z and allows to calculate F as F=VFTw.