q-Deformation of Rational Coxeter Friezes

Determine whether a suitable projective normalization makes the q-deformation constructed from q-rational points and q-cross-ratios depend only on a tame closed rational frieze, and determine whether the resulting array satisfies a natural q-diamond rule and preserves closure and periodicity.

Background

The paper establishes a relationship between q-cross-ratios, q-rational points, and q-deformed Coxeter friezes associated with integral Conway–Coxeter friezes. It then proposes extending this construction to tame closed friezes whose entries lie in the rationals.

The main obstruction is arithmetic and geometric: a rational polygon associated with a rational frieze is defined only up to PGL(2,Q), whereas the q-rational quantization map is equivariant only under the modular subgroup PGL(2,Z). Consequently, changing the projective representative may alter the resulting q-cross-ratios. The unresolved problem is to determine whether an appropriate normalization removes this dependence or whether additional arithmetic data are necessary, and whether the resulting rational q-frieze has the expected local rule, closure, and periodicity properties.

References

The question is therefore whether a suitable normalization can make this construction depend only on the rational frieze, or whether additional arithmetic data must be included. One must also determine whether the resulting array satisfies a natural $q$-diamond rule and whether closure and periodicity are preserved.

— The $q$-deformed cross-ratio: modular invariants and Coxeter friezes  (2609.10999 - Ovsienko, 10 Sep 2026) in Section 7, “Extensions and further questions,” subsection “q-deformation of rational friezes”