Ptolemy relations for punctured discs

TL;DR

This paper introduces frieze patterns for punctured discs, linking combinatorial matchings to cluster algebra variables, thereby expanding the understanding of algebraic structures associated with punctured surfaces.

Contribution

It constructs type D_N frieze patterns from triangulations of punctured discs and relates pattern entries to cluster variables in Fomin-Zelevinsky cluster algebras.

Findings

Frieze patterns are constructed from punctured disc triangulations.

Pattern entries correspond to matchings between vertices and triangles.

Entries can be interpreted as specialisations of cluster variables.

Abstract

We construct frieze patterns of type D_N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type D_N, we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster algebra.