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- Title
Dynamics of Fricke–Painlevé VI Surfaces.
- Authors
Planat, Michel; Chester, David; Irwin, Klee
- Abstract
The symmetries of a Riemann surface Σ ∖ { a i } with n punctures a i are encoded in its fundamental group π 1 (Σ) . Further structure may be described through representations (homomorphisms) of π 1 over a Lie group G as globalized by the character variety C = Hom (π 1 , G) / G . Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere Σ = S 2 (4) and the 'space-time-spin' group G = S L 2 (C) . In such a situation, C possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface V a , b , c , d (x , y , z) with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or P V I ); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of P V I . In this paper, we feature the parametric representation of some solutions of P V I : (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups f p encountered in TQC or DNA/RNA sequences are proposed.
- Subjects
RIEMANN surfaces; FUNDAMENTAL groups (Mathematics); HOMOMORPHISMS; QUANTUM computing; PAINLEVE equations
- Publication
Dynamics (2673-8716), 2024, Vol 4, Issue 1, p1
- ISSN
2673-8716
- Publication type
Article
- DOI
10.3390/dynamics4010001