# Possible values of winding numbers

Suppose that $$D$$ is a plane domain. Let $$N$$ be the set of all integers $$n$$ such that there is a closed, piecewise smooth curve $$\gamma$$ in $$D$$ whose winding number about the origin equals $$0$$: $$N:=\{n\in\mathbb{Z}\mid \textrm{there is a closed, piecewise smooth curve }\gamma \textrm{ in } D \textrm{ such that } n(\gamma ,0)=n\}.$$

(Here winding number $$n(\gamma ,0)$$ of a closed, piecewise smooth ($$C^1)$$ curve $$\gamma:[a,b]\rightarrow \mathbb{C}$$ is defined by $$n(\gamma,0):=\int_\gamma \frac{1}{z}dz.$$)

I was thinking about what the set $$N$$ look like in general. First I noticed that $$N$$ must be the set of the form $$k\mathbb{Z}$$, where $$k$$ is a nonnegative integer. After some drawing, I started to suspect that $$N$$ is either $$\{0\}$$ or $$\mathbb{Z}.$$ However, I can neither prove nor disprove it. Is it true that $$N=\{0\}$$ or $$\mathbb{Z}?$$ If so, why?

Any help is appreciated. Thanks in advance!