Interpret Farey-graph geometry for higher-dimensional SLk-friezes

Interpret the geometric notion of clockwise paths in the Farey graph for higher-dimensional SLk-friezes, in order to obtain a higher-dimensional analogue of the path-based classification of positive SL2-friezes and tilings.

Background

The paper generalizes the correspondence between SL2-tilings and paths in the Farey graph to tame SLk-tilings by replacing rational paths with algebraic paths of vectors in Zk. For k=2, the geometry of the Farey graph supports interpretations of positivity, periodicity, and frieze structures in terms of special paths, including clockwise paths.

For higher k, the authors introduce quiddity sequences and obtain partial positivity results for small values of k and n, but they do not identify a geometric counterpart to clockwise Farey paths. Establishing such an interpretation would provide a geometric framework for classifying positive higher-dimensional friezes and tilings.

References

On the other hand, the classification of positive SL2-friezes and tilings in terms of clockwise paths in the Farey graph obtained in [Sho23], relies heavily on the geometry of the Farey graph. It is not clear how to interpret this geometric notion in our setting, and in the case of SLk-friezes we introduce an algebraic notion of a quiddity sequence for higher values of k.

— $SL_k$-Tilings and Paths in $\mathbb{Z}^k$  (2504.01693 - Peterson et al., 2 Apr 2025) in Section 1, Introduction, p. 2