Prediction for (Sym^{(v)}, Sym^{(2)})-colored Hopf-link homology

Prove that the Poincaré polynomial of the (Sym^{(v)}, Sym^{(2)})-colored Khovanov–Rozansky homology of the Hopf link T(2,2) equals the rational function derived from Hilb^n({x^2 y^v=0},0), namely ((1−Q)(1−Q^2T^2) + (1−Q)(1+QT^2)Q^{v+1}T^{2v} + Q^{2(v+1)}T^{4v}) / ((1−Q)(1−Q^2T^2) ∏_{i=1}^{v}(1−Q^{i}T^{2(i−1)})).

Background

After computing the Poincaré polynomial on the Hilbert-scheme side for C = {x2 yv = 0}, the paper proposes that this matches the colored Hopf-link homology, providing a concrete prediction in the absence of an existing computation on the link-homology side.

Establishing this equality would both compute a new colored Hopf-link homology and furnish further evidence for the colored ORS conjecture in a non-reduced singular setting.

References

Conjecture The colored knot homology is \begin{align*} \frac{(1-Q)(1-Q2T2)+(1-Q)(1+QT2)Q{v+1}T{2v} + Q{2(v+1)}T{4v}{(1-Q)(1-Q2T2)\prod_{i=1}{v}(1-Q{i}T{2(i-1)})}. \end{align*}

— Hilbert scheme of points on non-reduced nodal curves  (2604.03111 - Luan, 3 Apr 2026) in Subsection: Poincaré polynomial of Hilb^n({x^2y^v=0},0)