Dynamics of Fricke–Painlevé VI Surfaces
Abstract
:1. Introduction
2. Materials and Methods
2.1. The Fricke–Painlevé VI Surface
The Cayley’s Nodal Cubic Surface
3. Algebraic Solutions of Painlevé VI Equation Mapping to Algebraic Surfaces
3.1. The Klein Solution
3.2. Solutions with Parameters
3.3. Solutions with Parameters
3.4. The Great Dodecahedron Solution
3.5. Three Extra Solutions Leading to an Algebraic FRICKE-Painlevé Surface
4. Further Algebraic Solutions of Painlevé VI Equation
4.1. The Icosahedral Solution 7
4.2. Dubrovin–Mazzocco Platonic Solutions
4.3. Solutions Related to the Valentiner Group
4.4. Two Extra Icosahedral Solutions
5. Discussion
5.1. Application to Character Varieties of Finitely Generated Groups
A Specific Example: (-Methyladenosine) Modifications
5.2. Perspectives
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Planat, M.; Chester, D.; Irwin, K. Dynamics of Fricke–Painlevé VI Surfaces. Dynamics 2024, 4, 1-13. https://doi.org/10.3390/dynamics4010001
Planat M, Chester D, Irwin K. Dynamics of Fricke–Painlevé VI Surfaces. Dynamics. 2024; 4(1):1-13. https://doi.org/10.3390/dynamics4010001
Chicago/Turabian StylePlanat, Michel, David Chester, and Klee Irwin. 2024. "Dynamics of Fricke–Painlevé VI Surfaces" Dynamics 4, no. 1: 1-13. https://doi.org/10.3390/dynamics4010001