Towards differential expansion for generic representations

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 $\mathcal{K}$ 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 $Z_X^R$ 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 ${\delta }^{\mathcal{K}}$, but by a whole non-decreasing sequence $\{k_j\}$, which is squeezed between two staircases, $\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{r}\left(\frac{j-1}{\delta +1}\right)\le k_j\le \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{r}\left(\frac{j-1}{\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{\Delta }}\right)$, where the upper bound involves Alexander polynomial $\mathrm{\Delta }(q^2)$ 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 $\delta +1=\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{\Delta }$ 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.