Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 502

Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice

A. I. Bobenko, T. Hoffmann, Yu. B. Suris

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to $-1$. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B"acklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogs to the analytic functions $z^alpha$ and $log z$.

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