Maps from S² to S²

When studying continuous maps f:Sn->Sn, i.e. from the n dimensional sphere to the n dimensional sphere, study the degree of the map. We define the degree of such a continuous map by how the continuous map acts on the homology groups of Sn. Hq(Sn) = Z if q = 0 or n and Hq(Sn) = 0 for other values of q. Since H0(X) only describes the number of connected components of a topological space X, the most interesting homology group of Sn is with q = n. A continuous map f:Sn->Sn induces in Hn a group homomorphism f*:Z->Z. This homomorphism is uniquely determined by f*(1) which we call deg(f). Now, polynomials in C become continuous maps from f:S2->S2 where S2 is the Riemann sphere of C. Its not hard to show that the degree of the polynomial is in fact equal to the degree deg(f) when the polynomial is viewed as a continuous function on the Riemann sphere. I wondered if this phenomenon could be seen. Below, I have drawn the image on the Riemann sphere of the annulus 0.9<|z|<0.1 in the complex plane under the action of a complex polynomial. The image below that is an animated linear interpolation between the annulus and its image.

Can you guess/determine the degree of the polynomial, i.e. the degree of the continuous map between Riemann spheres?

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