Independence of F''_H from the choice of auxiliary curves in virtual Heegaard diagrams

Demonstrate that the map F''_H defined on standardized virtual Heegaard diagrams (Σ_g, α, β, p_∞, γ) is independent of the choice of the auxiliary family of curves γ used to standardize the diagram, i.e., show that F''_H(Σ_g, α, β, p_∞, γ) is invariant under replacing γ by any other family producing a standardized virtual Heegaard diagram.

Background

The map F''_H is introduced as a computational tool for evaluating the chromatic spherical invariant via virtual Heegaard diagrams. It is shown to be invariant under various diagrammatic moves and choices (e.g., standard projection, certain isotopies, handle-slides, Reidemeister moves, and orientation changes), and the paper extends F''_H to broader classes via flattenings.

While the paper later derives independence with respect to the point and γ in the specific Heegaard-diagram setting by identifying F''_H with the chromatic invariant, the authors note that a general independence of F''_H from γ for standardized virtual Heegaard diagrams has not been proven.