KNTZ trick from arborescent calculus and the structure of differential expansion

Abstract

The recently suggested KNTZ trick completed the lasting search for exclusive Racah matrices $\bar{S}$ and $S$ for all rectangular representations and has a potential to help in the non-rectangular case as well. This was the last lacking insight about the structure of differential expansion of (rectangularly-)colored knot polynomials for twist knots -- and the resulting success is a spectacular achievement of modern knot theory in a classical field of representation theory, which was originally thought to be a tool for knot calculus but instead appeared to be its direct beneficiary. In this paper we explain that the KNTZ ansatz is actually a suggestion to convert the arborescent evolution matrix $\bar{S}{\bar{T}}^2\bar{S}$ into triangular form $\mathcal{B}$ and demonstrate how this works and what is the form of the old puzzles and miracles of the differential expansions from this perspective. The main new fully result is the conjecture for the triangular matrix $\mathcal{B}$ in the case of non-rectangular representation $[3,1]$. This paper does not simplify any calculations, but highlights the remaining problems, which one needs to overcome in order to {\it prove} that things really work. We believe that this discussion is also useful for further steps towards non-rectangular case and the related search of the gauge-invariant arborescent vertices. As an example we formulate a puzzling, still experimentally supported conjecture, that the study of twist knots only is sufficient to describe the shape of the differential expansion for all knots.

20 pages

References in corpus (14)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Refined Chern-Simons Theory and Topological String
• On colored HOMFLY polynomials for twist knots
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• Refined Chern-Simons Theory and Knot Homology
• On rectangular HOMFLY for twist knots
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Perspectives of differential expansion
• Extension of KNTZ trick to non-rectangular representations
• Universal Character and Large N Factorization in Topological Gauge/String Theory
• Kerov functions for composite representations and Macdonald ideal
• On exclusive Racah matrices $\bar{S}$ for rectangular representations

Cited by in corpus (9)

• Perspectives of differential expansion
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• Towards tangle calculus for Khovanov polynomials
• 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
• Defect and degree of the Alexander polynomial
• HOMFLY Polynomials of Pretzel Knots