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

arXiv comment14 pages

References in corpus (37)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• Knot Homology from Refined Chern-Simons Theory
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• On the Hecke algebras and the colored HOMFLY polynomial
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Evolution method and "differential hierarchy" of colored knot polynomials
• Colored HOMFLY polynomials for the pretzel knots and links
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Finalizing the proof of AGT relations with the help of the generalized Jack polynomials
• Refined Chern-Simons Theory and Topological String
• Chern-Simons Invariants of Torus Links
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Towards effective topological field theory for knots
• Differential hierarchy and additional grading of knot polynomials
• On colored HOMFLY polynomials for twist knots
• Tabulating knot polynomials for arborescent knots
• Colored Kauffman Homology and Super-A-polynomials
• Colored HOMFLY polynomials that distinguish mutant knots
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• On the defect and stability of differential expansion
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• CFT exercises for the needs of AGT
• Refined Chern-Simons Theory and Knot Homology
• On rectangular HOMFLY for twist knots
• Differential expansion for link polynomials
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Are there p-adic knot invariants?
• Extension of KNTZ trick to non-rectangular representations
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• HOMFLY for twist knots and exclusive Racah matrices in representation [333]
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Pentad and triangular structures behind the Racah matrices

Cited by in corpus (12)

• Distinguishing Mutant Knots
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• A new symmetry of the colored Alexander polynomial
• Towards tangle calculus for Khovanov polynomials
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Algebra of quantum $\mathcal{C}$-polynomials
• Evolution properties of the knot's defect
• Towards the theory of Yangians
• Defect and degree of the Alexander polynomial
• On geometric bases for quantum A-polynomials of knots