There is a formula for the simple closed curve $\gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(\gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the closed curve $\gamma(t)$. This is just a direct result from the argument principle.
Now I want to ask what if the closed curve $\gamma(t)$ is NOT simple anymore which means it may have self-intersections. What will the winding number $w(p(\gamma),(0,0))$ be? I think it should be still related to the roots of $p(z)$