Towards differential expansion for generic representations

Abstract

We report the first results of phenomenological analysis, based on extended set of explicitly calculated colored HOMFLY-PT polynomials in \cite{knotebook}. They concern differential expansion (DE), which separates the dependencies on the knot ${\cal K}$ and the representation $R$. %of colored HOMFLY-PT polynomials. DE is well understood for symmetric and rectangular representations, while for non-rectangular $R$ even the knot-independent basis $Z_R^X$ of DE is not known. We concentrate on two issues: the possible derivation of DE and on improved characteristic of its defect. We discuss how evolution families -- twist knots, double braids and 3-pretzels -- can be used to find DE by a peculiar triangular transformation of evolution coefficients. Once $Z^R_X$ are found for particular family, they are expected to stay basically the same for arbitrary knots -- modulo controllable reductions, characterized by the defect of DE, which puts knots into different universality classes. Contrary to original expectation, these classes appear labeled not by a single number $δ^{\cal K}$, but by a whole non-decreasing sequence $\{k_j\}$, which is squeezed between two staircases, ${\rm entier}\left(\frac{j-1}{δ+1}\right)\leq k_j\leq{\rm entier}\left(\frac{j-1}{{\rm deg}\,Δ}\right)$, where the upper bound involves Alexander polynomial $Δ(q^2)$ and follows from the continuation of the Melvin--Morton--Rozansky-Kashaev limit to $A=q^{-m}$ with $m>0$, when the limit gives a {\it positive} power of Alexander. The old assumption $δ+1={\rm deg}\,Δ$ is oversimplified and fails for 19 non-alternating knots with 11 and 12 crossings. Last, we list the differences of non-rectangularly colored HOMFLY for various mutant pairs to demonstrate their universality (knot-independence) and nice consistency with the spirit of DE.