Colored Hopf-link homology for (Sym^{(v)}, Sym^{(2)}) is unknown

Determine the Poincaré polynomial of the (Sym^{(v)}, Sym^{(2)})-colored Khovanov–Rozansky homology of the Hopf link T(2,2), in order to enable verification of the ORS conjecture for the non-reduced singular curve C = {x^2 y^v = 0}.

Background

For C = {x2 yv = 0}, the associated link is the Hopf link with one component colored by a row of v boxes and the other by a row of 2 boxes. The paper determines the Poincaré polynomial of the Hilbert scheme side but cannot compare it to the link side because the latter is not yet known.

Computing this colored Hopf-link homology would allow a direct test of the colored ORS conjecture in the first non-reduced singular case beyond u=1.

References

Because the Poincar e polynomial for the Hopf link is not known in this case, we cannot yet verify the ORS conjecture.

— Hilbert scheme of points on non-reduced nodal curves  (2604.03111 - Luan, 3 Apr 2026) in Subsubsection: Affine paving of Hilb^{n}({x^2y^v=0},0) (Results of this paper)