2-frieze patterns and the cluster structure of the space of polygons

TL;DR

This paper explores the space of 2-frieze patterns, generalizing classical frieze patterns, and reveals its structure as a cluster manifold related to moduli spaces of polygons and genus 0 curves.

Contribution

It introduces a new class of 2-frieze patterns, establishes their geometric realization, and demonstrates their cluster manifold structure.

Findings

The space of 2-frieze patterns is a cluster manifold.

Connections between 2-frieze patterns and moduli spaces of polygons.

Analysis of algebraic and arithmetic properties of these patterns.

Abstract

We study the space of 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.