Classification of quad-equations on a cuboctahedron

TL;DR

This paper classifies consistent polynomial systems associated with a cuboctahedron's faces, leading to partial difference equations on a lattice, motivated by discrete Painlevé equations and their tau-functions.

Contribution

It introduces a classification of polynomials linked to a cuboctahedron that produce consistent systems of difference equations, connecting geometric and integrable systems.

Findings

Classified polynomials associated with cuboctahedron faces.

Derived partial difference equations on a face-centred cubic lattice.

Linked results to discrete Painlevé tau-functions.

Abstract

In this paper, we consider polynomials associated with faces and internal quadrilaterals of a cuboctahedron and classify them under the requirement that they are consistent. These polynomials give rise to a system of partial difference equations on a face-centred cubic lattice. Our results were motivated by $\tau$-functions related to discrete Painlev\'e equations.