On maximal Dynkin friezes

Abstract

The maximal entries of Dynkin friezes over the positive integers have recently been determined for all finite Dynkin types except $B_n$ and $D_n$. In this note, we explicitly construct large positive integral points on affine cluster varieties of type $B_n$ (resp. $D_n$), giving rise to friezes of types $B_n$ (resp. $D_n$) over the positive integers with largest entries $F_{n+1}{F}_{n+2}-1$ (resp. $F_n{F}_{n+1}-1$) where $F_k$ is the $k$-th Fibonacci number. We conjecture that these are the maximal possible entries for their respective Dynkin types.

15 pages