# When is the winding number undefined

Let $$\phi_R(t)=R(\cos 2t + i\sin 2t)$$ be the closed circle of radius $$R \geq 0$$ going twice around the origin. Consider the closed curve $$P_R(t)=p(\phi_R(t))$$, where $$p(z)$$ is the polynomial $$z^3+z^2-2z-2$$. What is the smallest $$R>0$$ for which the winding number of $$P_R(t)$$ is undefined?

• Winding number with respect to which point? – Christian Blatter Nov 10 '18 at 14:24
• If $R^2=2$ the curve degenerates to a point. – Michael Hoppe Nov 10 '18 at 15:09