The motivic–DAHA coincidence conjecture for colored algebraic links

Let $\mathcal{R}$ be a plane-curve-singularity ring with branches indexed by $1\le i\le\kappa$, and let $\sigma=(c_1\ge\cdots\ge c_\kappa>0)$. Define the motivic superpolynomial

$$\mathcal{H}^{mot}_{\sigma}=\sum_M t^{{\text{\rm dim}}_{\mathbb C}\,(\Omega/M)}\prod_{j=c_1}^{r\!k_q(M)-1}(1+aq^j),$$

where the sum is over standard modules $M\subset\Omega=\bigoplus_i\mathcal{O}_i^{c_i}$ and $r\!k_q(M)={\text{\rm dim}}_{\mathbb C}\,_{\mathbf{F}_q}(M/\mathfrak m M)$. The motivic–DAHA coincidence conjecture. The polynomial $\mathcal{H}^{mot}_{\sigma}$ depends polynomially on $q,t,a$ and coincides with the DAHA superpolynomial $\widehat{\mathcal{H}}^{\lambda}(q,t,a)$ of the corresponding algebraic link colored by $\{\lambda\}=\{c_1\omega_1,\ldots,c_\kappa\omega_1\}$, with only pure rows. Consequently, it is a topological invariant of the link. In the uncolored case $\sigma=(1,\ldots,1)$, it satisfies $(qt)^\delta\mathcal{H}^{mot}(1/t,1/q,a)=\mathcal{H}^{mot}(q,t,a)$.