A note on friezes of type \(\varLambda_p\) (Q2182199)

In [Acta Arith. 18, 297--310 (1971; Zbl 0217.18101)], \textit{H. S. M. Coxeter} introduced and studied frieze patterns, and proved some surprising properties like their periodicity. Two years later, in collaboration with Conway, he proved by elementary methods, a number of beautiful results including the fact that there is a natural bijection between integral frieze patterns and triangulations of polygons [\textit{J. H. Conway} and \textit{H. S. M. Coxeter}, Math. Gaz. 57, 87--94 (1973; Zbl 0285.05028); ibid. 57, 175--183 (1973; Zbl 0288.05021)]. Frieze patterns with more general, non-integral entries have also come to be studied by many mathematicians revealing unexpected connections between several areas. Related to a frieze pattern is the notion of a quiddity sequence (defined by Coxeter) and a positive integer called its width. A recent work by \textit{T. Holm} and \textit{P. Jørgensen} [``A \(p\)-angulated generalisation of Conway and Coxeter's theorem on frieze patterns'', Int. Math. Res. Not. 2020, Nr. 1, 71--90 (2020; \url{doi:10.1093/imrn/rny020})] considers the problem of \(p\)-angulation (dissecting a polygon into \(p\)-gons) for a positive integer \(p \geq 3\). They look at frieze patterns whose quiddity sequence consists of integral multiples of \(2 \cos (\pi/p)\) for some fixed integer \(p \geq 3\). They call such frieze patterns to be of type \(\Lambda_p\). Note that frieze patterns of type \(\Lambda_3\) have an integral quiddity sequence. The authors prove that there is a bijection between frieze patterns of type \(\Lambda_p\) and width \(n\) and \(p\)-angulations of an \((n+3)\)-gon; they use the theory of Hecke groups. In that paper, Holm and Jørgensen [loc. cit.] raise the question of usefully characterizing frieze patterns of type \(\Lambda_p\) for \(p \geq 4\) like the one for \(\Lambda_3\). In the paper under review, the author partially solves this for \(p=4\) and \(p=6\). More precisely, he proves: Let \(p=4\) or \(p=6\). For every \(p\)-angulation \(D\) of a polygon \(P\), there exist exactly two triangulations such that their Conway-Coxeter friezes coincide with the frieze of type \(\Lambda_p\) associated to \(D\) in every second row. Further, a similar result cannot hold for any other \(p\).