The DAHA-Jones polynomial conjecture

Let $K=K_{r,s}$ be a torus knot with $r,s\in\mathbf{Z}$, let $R$ be a root system, and let $b\in P_+$ be a dominant weight. Define

$$J\!D_{r,s}^{R}(b;q,t)=J\!D_{r,s}(b;q,t)=\{\,\widetilde{\gamma}_{r,s}(P_b)/P_b(q^{-\rho_k})\,\}.$$

Here $P_b$ is the symmetric Macdonald polynomial and $\widetilde{\gamma}_{r,s}$ is a braid-group representative of the knot. DAHA-Jones polynomial conjecture. The tilde-normalized polynomial $\widetilde{J\!D}_{r,s}(b;q,t)$ is independent of the chosen representative $\gamma_{r,s}$, is a polynomial in $q,t_\nu$, and satisfies

$$\widetilde{J\!D}_{r,s}(b;q,t\mapsto q)=\widetilde{\mathcal{J}}_{r,s}(b;q)$$

when $k_\nu=1$, where $\mathcal{J}_{r,s}(b;q)$ is the Jones polynomial associated with the root system $\widetilde R$ and highest weight $b$, normalized by $\mathcal{J}(\text{unknot})=1$. This conjecture asserts both well-definedness of the DAHA-Jones invariant and its specialization to the quantum-group Jones polynomial; the coincidence is verified in several types, including $A_n$, but is not established in full generality.