The $q$-deformed cross-ratio: modular invariants and Coxeter friezes

We introduce and study a scalar $q$-deformation of the cross-ratio on ${\mathbb{P}}^1(\mathbb{Q})$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.