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I've been given the following loop $\gamma$, which clearly divides the complex plain into 5 domains. For each of these domains, I have been asked to find the winding number of $a$ around $\gamma$, where $a$ is a point in the domain.

Since I haven't been given any further information of the loop, I don't think I can use the formula $$n(\gamma,a) = \frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z-a}$$ now intuitively I know that the winding numbers of the domain outside the loop is $0$, the domain in the centre is, I'm guessing, $2$ and for the remaining domains is $1$. But nowhere have I been able to find a good explanation for winding number apart from the aforementioned formula (and the proof that it will be an integer)[Resources consulted: Couple of Lecture Notes online, this website, and books by Lang, Ahlfors and Bak-Newman]. Can someone please give me an explanation on how I can find the winding number in situations like these? I haven't yet been taught the Cauchy Integral Formula.

José Carlos Santos
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Naweed G. Seldon
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    Is https://math.stackexchange.com/questions/703695/crazy-calculation-for-winding-numbers/704176#704176 of some help? – Michael Hoppe Sep 30 '18 at 14:41
  • This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results. – minmax Sep 30 '18 at 15:01

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You left out a vital piece of information: that little arrow on the left, pointing down. It tells you in which direction the loop is travelled (only once, I assume). Therefore, the winding number of each point of the three lobes is $1$ and in the central region it is equal to $2$. I suggest that you read this intuitive description of the winding number.

José Carlos Santos
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