Combinatorial construction of the master polynomial

Construct a direct combinatorial model for the trivariate master polynomial H_{R,n}(q,t,u) of a plane curve germ, including the third grading that is invisible in the two known specializations.

Background

The proposed master polynomial is characterized conjecturally by several identities, but the paper does not provide an intrinsic construction. The missing third variable is intended to encode the perverse grading and to refine the two-variable rational q,t-Catalan-type series. The authors explicitly identify the absence of a direct combinatorial construction as an open issue.

References

In higher rank, a direct combinatorial construction remains open.

— Quot scheme of points on torus knot singularities  (2608.16086 - Huang et al., 17 Aug 2026) in Section 1, subsection “Speculations on a trivariate refinement”