# Change in argument of sum of complex numbers along a closed path

Let $$a(t)$$ and $$b(t)$$ be complex variables that travel along closed paths that end up at their original points, i.e. $$a(0)=a(T)=a_0$$ and $$b(0)=b(T)=b_0$$, with $$\Delta Arg(a) = \Delta Arg(b) = 0$$ over the paths. Is it possible for their sum $$c(t) = a(t) + b(t)$$ to have $$\Delta Arg(a) = 2\pi k$$ where $$k$$ is a non-zero integer? Any example? Also, let's assume $$a(t) \neq 0, b(t) \neq 0$$, and $$c(t) \neq 0$$ for any $$0 \leq t \leq T$$.