Frieze patterns and combinatorics of curve singularities

TL;DR

This paper explores the relationship between Conway-Coxeter frieze patterns and the resolution data of complex curve singularities, establishing a bijection and interpreting frieze entries through representation theory.

Contribution

It introduces a novel bijection between resolution graphs of Newton non-degenerate plane curve singularities and Conway-Coxeter friezes, linking combinatorics and algebraic geometry.

Findings

Established a bijection between resolution graphs and frieze patterns.

Interpreted frieze entries via representation theory.

Connected mutation concepts to curve resolutions.

Abstract

We study the connection between Conway-Coxeter frieze patterns and the data of the minimal resolution of a complex curve singularity: using Popescu-Pampu's notion of the lotus of a singularity, we describe a bijection between the dual resolution graphs of Newton non-degenerate plane curve singularities and Conway-Coxeter friezes. We use representation theoretic reduction methods to interpret some of the entries of the frieze coming from the partial resolutions of the corresponding curve singularity. Finally, we translate the notion of mutation, coming from cluster combinatorics, to resolutions of plane complex curves.