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Characterize the epichristoffel word in each conjugacy class

Determine a general characterization of the epichristoffel word for each conjugacy class of finite words over alphabets of size at least three, i.e., identify the unique Lyndon representative within every conjugacy class that belongs to an epichristoffel class, by an explicit and general method.

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Background

Epichristoffel words generalize Christoffel words to alphabets with three or more letters via episturmian morphisms. While many properties of Christoffel words carry over, several do not, and Paquin highlighted specific gaps. The authors introduce epichristoffel trees to partially address these issues, showing how to obtain the epichristoffel representative for classes that appear on the tree (e.g., Theorem 5.1 and Example 5.3). However, not every epichristoffel word appears on a given tree, so a complete, uniform characterization across all conjugacy classes remains an open challenge as originally posed.

References

Although epichristoffel words share many of the same properties as Christoffel words, Genevieve raised some open problems regarding epichristoffel words. These include the ability to characterize the epichristoffel word of each conjugacy class, whether epichristoffel words satisfy a type of balanced property, and whether there is an epichristoffel word of any length over a k-letter alphabet for a fixed k ≥ 3.

— On a Generalization of the Christoffel Tree: Epichristoffel Trees (2507.15313 - Krishnamoorthy et al., 21 Jul 2025) in Section 1. Introduction