Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing

TL;DR

This paper extends Coxeter friezes to a non-commutative setting, providing formulas for determinants, Laurent phenomena, and methods for gluing friezes, advancing the understanding of non-commutative frieze structures.

Contribution

It introduces a non-commutative generalisation of Coxeter friezes, including determinant formulas, Laurent phenomenon expressions, and frieze gluing techniques.

Findings

Derived a formula for non-commutative frieze determinants

Established a T-path formula for Laurent phenomenon in non-commutative context

Developed methods for gluing non-commutative friezes

Abstract

This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a $T$-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Canakci and Jorgensen.