Uniqueness of quad mesh combinatorics given singularity indices

Prove that, for a closed triangulated surface, the combinatorial structure of the quad mesh implied by a grid-preserving (seamless with integer translations) map from the surface to R^2 is uniquely determined by the set of indices of its singularities. For surfaces with boundary, establish the corresponding uniqueness under boundary-wise index prescriptions that mirror the closed-surface condition.

Background

The paper studies quad mesh extraction from grid-preserving maps, focusing on how non-ideal inputs affect the combinatorics of the resulting mesh. The authors analyze how local perturbations in the map (noise) can alter the segmentation through operations such as saddle lifts, potentially changing the quad mesh combinatorial structure without necessarily altering singularity indices.

Within this context, they propose a conjecture that the quad mesh combinatorics are uniquely determined by the indices of singularities, at least on closed surfaces, with analogous conditions required on boundaries for open surfaces. This conjecture aims to disambiguate the multiple combinatorial outcomes that can arise from different saddle lifts and map deformations, linking the mesh structure directly to singularity index data.