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 $0 \leq t \leq T$.