The higher-order determination conjecture for Bar-Natan–van der Veen polynomials

Let $K$ be a 0-framed knot. The universal invariant $Z_\mathbb{D}(K)$ is associated with the Alexander polynomial $\Delta_K$ and polynomials $\rho_K^{i,j}$, where

$$\rho_K^{i,0}\in\mathbb{Q}[t,t^{-1}],\qquad i>0.$$

Higher-order determination conjecture. The value of $Z_\mathbb{D}(K)$ is completely determined by the Alexander polynomial $\Delta_K$ and the family of polynomials $\rho_K^{i,0}$, $i>0$. More precisely, if $j>0$, the value $\rho_K^{i,j}$ can be written in terms of $\Delta_K$, the polynomials $\rho_K^{i,0}$ with $i<j$, and derivatives of these.

This extends the known determination of the first terms from $\Delta_K$, $\rho_K^{1,0}$, and $\rho_K^{2,0}$; the statement is proposed as an open problem for higher-order terms.