preprints.org > computer science and mathematics > geometry and topology > doi: 10.20944/preprints202312.0114.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

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)

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.

isomonodromic deformation; Painlev\'e VI; $SL(2,\mathbb{C})$ character variety; algebraic surfaces; DNA/RNA

Computer Science and Mathematics, Geometry and Topology

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