Zeros and poles

The three solid dots are zeros and the two rings are poles; drag them and the function changes with them. Walk once anticlockwise around a zero and the colours go round once, in the order of the wheel at the bottom right; around a pole they go round the other way. Put two zeros together and the colours wind twice; drop a zero onto a pole and the two cancel, and the colours there stop turning. Walk round the edge of the picture with all five points inside, and the colours turn just once: there is one more zero than poles.