The basilica mated with the basilica

The basilica polynomial is the polynomial F(z)=z2-1.
Critical points of F: 0, ∞
Dynamic portrait: 0 -2-> -1 -1-> 0, ∞ -2-> ∞

There exists an open topological disk in the Riemann sphere containing ∞ stabilized by F on which the action of F is equivalent to the action of z↦ z2 on the open unit disk. Let D be the complement of this open topological disk in the Riemann sphere. Then F stabilizes D. We take two copies of D and identify their boundaries in the straightforward way so that two copies of F combine to obtain the mating f of F with itself.

Critical portrait of f: a -2-> b -1-> a, c -2-> d -1-> c

The diagram for f

According to Kelsey and Lodges' paper ``Quadratic Thurston maps with few postcritical points'' Geom. Dedicata 201 (2019), 33-55, especially Tables 2 and 4, every Thurston map with this dynamic portrait is obstructed. Furthermore these maps are all Thurston equivalent to twists of any one such map, say f, by powers of a Dehn twist about the obstruction. The f-pullback of the obstruction is homotopic to the obstruction, rel Pf={a,b,c,d}, and the same is true for another homotopy class of curves in S2 - Pf, namely, the homotopy class of the equator of the mating. This property determines the Thurston equivalence class of f among these Thurston equivalence classes. Now one verifies that the following NET map presentation diagram yields a NET map with the same dynamic portrait as f. It has an obstruction with slope ∞ and an equator with slope 0. It must be Thurston equivalent to f.

Input and output files