Homeomorphisms between limbs of the Mandelbrot set

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of equal denominator. Induced by these homeomorphisms and complex conjugation, we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinatorial level relating corresponding external arguments. Assuming local connectivity of the Mandelbrot set we may conclude that the constructed homeomorphisms between limbs are compatible with the embeddings of the limbs in the plane. As usual we plough in the dynamical planes and harvest in the parameter space.