Is Wikipedia's "Pochhammer contour" wrong? It's winding number is NOT zero around some points. Wikipedia claims in this article, in the first section, it reads

Homologous to zero but not homotopic to zero Within the doubly
  punctured plane this curve is homologous to zero but not homotopic to
  zero. Its winding number about any point is 0 despite the fact that
  within the doubly punctured plane it cannot be shrunk to a single
  point.

This is the image of wikipedia of the curve.
This is true for the red and green points, however, the winding number changes by 1 when you cross the curve, so it's not zero in the other 2 bounded connected components. Also, you can readily "visually compute" the winding number of such a simple curve by looking how many turns around the point does, and for the portions I just described, it obviously doesn't do zero.
Side question: Are there curves with 0 winding number everywhere, not constant, not trivially going trought themselves?
Now, obviosuly, there exist curves homotopic to zero. Are there homologous to zero (my definition of this is without points with no-null winding number. I suppose tottaly impossible, because if I cross the curve, the winding number changes, so how can that be? The curve probably should travel over the same points twice, such that when I cross it, it changes twice the winding number to stay the same.
 A: The winding number of a curve needs to be computed with the entire curve and a chosen point in mind. This curve does indeed have winding number zero. To see this, pick either the red or green point. Begin at one of the self-intersections of the curve, and follow the curve with its given orientation. When you make a full counterclockwise turn (hence returning to the starting point), mark $+1$; when you make a full clockwise turn and return to the start, mark $-1$. You will then see that the Pochhammer curve turns once counterclockwise, and once clockwise, around each point. Therefore the winding number is $+1-1 = 0$.
This also furnishes an example for your follow-up question: as claimed in the article, the Pochhammer curve is a curve homologous to zero in the doubly punctured plane. The existence of closed curves with nontrivial homotopy or homology is a topological question, and is not well-founded without specifying a domain. In the case of the entire complex plane, every closed curve is homotopic, and therefore homologous, to zero. In the singly punctured plane, a curve is homologous to zero if and only if it is homotopic to zero, which comes down to the algebraic topology fact that $S^1$ has $\mathbb{Z}$ as its fundamental group, which coincides with its first homology. In the doubly punctured plane the fundamental group (free group on two generators) no longer coincides with the first homology ($\mathbb{Z}^2$), and this is reflected in the existence of curves in this space that are homologous, but not homotopic, to zero.
A: As a belated remark:
Bear in mind that a curve $C$ being "homologous to zero" and "homotopic to zero" must be understood with respect to some region $\Omega$ it lies in. $C$ is homologous to zero with respect to $\Omega$ if the winding number $\chi(C;z)$ is zero for all $z \not \in \Omega$.
So, for example, the unit circle is nullhomologous as a curve in $\mathbb{C}$, but not as a curve in $\mathbb{C} \backslash \{0\}$.
In your example it is stated that the curve is nullhomologous within the doubly punctured plane. That means the winding numbers at the two puncture points are zero. It does not claim anything about the other points.
