Friezes, weak friezes, and T-paths.

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T -paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T -paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T -path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued. [ABSTRACT FROM AUTHOR]