Wild frieze patterns over the integers

We study frieze patterns over the integers that are allowed to have wild entries. We introduce the quiddity number as a new invariant. The quiddity number is then used to classify strongly connected components of the directed graph ${\mathrm{\Gamma }}_{2,n}(\mathbb{Z})$. Furthermore, we show that every finite simple directed graph arises as an induced subgraph of a directed graph ${\mathrm{\Gamma }}_{2,n}(\mathbb{Z})$ for $n$ sufficiently large.