Global topology and injectivity of the Klein combination locus

Prove that the Klein combination locus is connected, that its component cut along the interval (-1,1) and with the modular Mandelbrot set removed is simply connected, and that the canonical map Ψ extends uniquely and injectively to this cut domain.

Background

The paper introduces a component of the Klein combination locus containing the disc D(4,3), removes the real interval (-1,1), and considers the resulting cut domain outside the modular Mandelbrot set. The conjecture asserts a global topological description of the locus and a strong global extension and injectivity property for Ψ, analogous to the role of the external parameterization for the classical Mandelbrot set.

References

$$ is connected, that is, $0=$; $_0{cut}\setminus M$ is simply-connected; $\Psi$ extends uniquely from ${\mathcal D}\setminusM_$ to $0{cut}\setminus M$, and this extension is injective on $0{cut}\setminus M$.

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Conjecture 3.??, Section 3.2 “The global structure of $$”

However we do not know of any algorithm to plot the pull-backs under $\Psi$ of the lines making up a standard tessellation of ${\mathbb H}$, the analogues of the Douady-Hubbard {\it parameter rays} for quadratic polynomials.

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Appendix I, Section 4 “Plotting the tessellation”