Factorization of Frieze Patterns

TL;DR

This paper investigates the algebraic properties of quiddity cycles in frieze patterns, revealing that their sum operation is neither commutative nor associative and that unique factorization does not hold.

Contribution

It analyzes the algebraic structure of quiddity cycles, showing the non-commutativity and non-associativity of their sum, and demonstrates the non-uniqueness of their irreducible decompositions.

Findings

Sum operation is not commutative.

Sum operation is not associative.

Decomposition into irreducible factors is not unique.

Abstract

In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the negative by giving counterexamples. We conclude that even under stronger assumptions, there is no canonical decomposition.