Search Results (4)

The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{a_i\right\}$ with n punctures $a_i$ are encoded in its fundamental group ${\pi }_1\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_1$ over a Lie group G as globalized by the character variety $\mathcal{C}=\mathrm{Hom}({\pi }_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere $\mathsf{\Sigma }=S_2^{\left(4\right)}$ and the ‘space-time-spin’ group $G=S{L}_2\left(\mathbb{C}\right)$. In such a situation, $\mathcal{C}$ 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 $P_{VI}$); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of $P_{VI}$. In this paper, we feature 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. Full article

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group $\pi$, we compute the $SL(2,\mathbb{C})$ character variety, where $SL(2,\mathbb{C})$ is simultaneously a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when $\pi$ looks similar to a free group $F_r$ ($r=1$ to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis $\mathcal{G}$ of the variety. A surface with simple singularities (such as the well known Cayley cubic) within $\mathcal{G}$ is a signature of a potential disease even when $\pi$ looks similar to a free group $F_r$ in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in $\{A,T/U,G,C\}$. We produce a few tables of human TFs and miRNAs showing that a disease may occur when either $\pi$ is away from a free group or $\mathcal{G}$ contains surfaces with isolated singularities. Full article

We recently proposed that topological quantum computing might be based on $SL(2,\mathbb{C})$ representations of the fundamental group ${\pi }_1(S^3\backslash K)$ for the complement of a link K in the three-sphere. The restriction to links whose associated $SL(2,\mathbb{C})$ character variety $\mathcal{V}$ contains a Fricke surface ${\kappa }_d=xyz-x^2-y^2-z^2+d$ is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link $L2a1$, one of the three arithmetic two-bridge links (the Whitehead link $5_1^2$, the Berge link $6_2^2$ or the double-eight link $6_3^2$) or the link $7_3^2$, the $\mathcal{V}$ for those links contains the reducible component ${\kappa }_4$, the so-called Cayley cubic. In addition, the $\mathcal{V}$ for the latter two links contains the irreducible component ${\kappa }_3$, or ${\kappa }_2$, respectively. Taking $\rho$ to be a representation with character ${\kappa }_d$ ($d<4$), with $\left|x\right|,\left|y\right|,\left|z\right|\le 2$, then $\rho \left({\pi }_1\right)$ fixes a unique point in the hyperbolic space ${\mathcal{H}}_3$ and is a conjugate to a $SU\left(2\right)$ representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example. Full article

It is shown that the representation theory of some finitely presented groups thanks to their $S{L}_2\left(\mathbb{C}\right)$ character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of $S{L}_2\left(\mathbb{C}\right)$ character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface $f_H$ (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers ${\tilde{E}}_6$ and ${\tilde{D}}_4$ contain $f_H$. A surface birationally equivalent to a $K_3$ surface is another compound of their character varieties. Full article