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Dynamics of Fricke-Painlevé VI surfaces
Version 1
: Received: 2 December 2023 / Approved: 4 December 2023 / Online: 4 December 2023 (03:27:06 CET)
Version 2 : Received: 22 December 2023 / Approved: 25 December 2023 / Online: 25 December 2023 (05:36:30 CET)
Version 2 : Received: 22 December 2023 / Approved: 25 December 2023 / Online: 25 December 2023 (05:36:30 CET)
A peer-reviewed article of this Preprint also exists.
Planat, M.; Chester, D.; Irwin, K. Dynamics of Fricke–Painlevé VI Surfaces. Dynamics 2024, 4, 1-13. Planat, M.; Chester, D.; Irwin, K. Dynamics of Fricke–Painlevé VI Surfaces. Dynamics 2024, 4, 1-13.
Abstract
The symmetries of a Riemann surface $\Sigma \setminus \{a_i\}$ with $n$ punctures $a_i$ are encoded in its fundamental group $\pi_1(\Sigma)$. Further structure may be described through representations (homomorphisms) of $\pi_1$ over a Lie group $G$ as globalized by the character variety $\mathcal{C}=\mbox{Hom} (\pi_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the $4$-punctured Riemann sphere $\Sigma=S_2^{(4)}$ and the \lq space-time-spin' group $G=SL_2(\mathbb{C})$. In such a situation, $\mathcal{C}$ possesses remarkable properties
(i) a representation is described by a $3$-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with $3$ variables and $4$ parameters, (ii) the automorphisms of the surface satisfy the dynamical (non linear and transcendental) Painlev\'e VI equation (or $P_{VI}$), (iii) there exists a finite set of $1$ (\mbox{Cayley-Picard})+$3$ (\mbox{continuous platonic})+$45$ (\mbox{icosahedral}) solutions of $P_{VI}$.
In this paper we feature on the parametric representation of some solutions of $P_{VI}$, (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.
Keywords
isomonodromic deformation; Painlev\'e VI; $SL(2,\mathbb{C})$ character variety, algebraic surfaces; DNA/RNA
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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