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

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

2001-04-25

arxiv.org/abs/math/0104244v1

Intern. Math. Research Notices, 2002, No 3, p.111-164

Mathematics

Complex Variables

43 pages, 13 figures

Scientific paper

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 analogons to the analytic functions $z^\alpha$ and $\log z$.

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