Quantum continuants, quantum rotundus and triangulations of annuli

TL;DR

This paper explores quantum analogues of classical mathematical objects called continuants and rotundi, providing combinatorial models involving polygon and annulus triangulations, and linking them to path and loop generating functions.

Contribution

It introduces quantum continuants and rotundi with combinatorial interpretations related to triangulations, extending classical concepts into the quantum domain.

Findings

Quantum continuants are coarea-generating functions of paths in triangulated polygons.

Quantum rotundi are (co)area-generating functions of closed loops on triangulated annuli.

The work connects quantum polynomials to combinatorial models involving polygon and annulus triangulations.

Abstract

We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.