Michel Planat, David Chester, Klee Irwin (2023)

The symmetries of a Riemann surface Σ{ai} with n punctures ai are encoded in its fundamental group π1(Σ). Further structure may be described through representations (homomorphisms) of π1 over a Lie group G as globalized by the character variety C=Hom(π1,G)/G. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the 4-punctured Riemann sphere Σ=S(4)2 and the \lq space-time-spin’ group G=SL2(C). In such a situation, C possesses remarkable properties (i) a representation is described by a 3-dimensional cubic surface Va,b,c,d(x,y,z) with 3 variables and 4 parameters, (ii) the automorphisms of the surface satisfy the dynamical (non linear and transcendental) Painlev\’e VI equation (or PVI), (iii) there exists a finite set of 1 (\mbox{Cayley-Picard})+3 (\mbox{continuous platonic})+45 (\mbox{icosahedral}) solutions of PVI. In this paper we feature on the parametric representation of some solutions of PVI, (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups fp encountered in TQC or DNA/RNA sequences are proposed.