Generalize remaining Farey-graph-based correspondences to higher k

Generalize the results for SL2-tilings and friezes that rely on the geometry of the Farey graph to SLk-tilings and friezes for higher values of k.

Background

The paper proves a higher-dimensional analogue of Short’s principal bijection between pairs of paths and SL2-tilings. However, Short’s work and related results also establish additional correspondences by imposing geometric restrictions on Farey-graph paths, including interpretations connected with friezes and positivity.

Because the vector-path framework in Zk lacks an established analogue of Farey-graph geometry, the paper does not generalize all of those additional correspondences. The unresolved task is to find higher-dimensional structures or methods that recover these results beyond the principal bijection.

References

In this paper, we obtain a generalization of this result for all k ≥ 2. Moreover, the geometry of the Farey graph allows to establish further bijections by placing certain restrictions on paths. In particular, this leads to the classification of friezes as well as positivity for friezes and tiling in terms of paths. While the main result of [Sho23] and several others are extended here, it is not clear how to generalize all of them as we lack the connection to geometry.

— $SL_k$-Tilings and Paths in $\mathbb{Z}^k$  (2504.01693 - Peterson et al., 2 Apr 2025) in Section 2.3, immediately following Theorem 2.23