Fricke Topological Qubits

We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for complement a link K in three-sphere. The restriction to links whose associated character variety V contains Fricke surface κd=xyz−x2−y2−z2+d is desirable due connection spaces elementary topology. Taking as Hopf L2a1, one three arithmetic two-bridge (the Whitehead 512, Berge 622 or double-eight 632) 732, those reducible component κ4, so-called Cayley cubic. In addition, latter two irreducible κ3, κ2, respectively. ρ representation with κd (d<4), |x|,|y|,|z|≤2, then ρ(π1) fixes unique point hyperbolic space H3 and conjugate SU(2) (a qubit). Even though details physical implementation remain open, more generally, we show may developed from view three-bridge links, topology four-punctured sphere Painlevé VI equation. 0-surgery circles Borromean rings L6a4 taken an example.