If you have a picture of a contour you can find the winding number by using the rule 'add one if you cross the contour while it's going down' and 'subtract one if you cross it when it's going up'. In some cases however it seems I can get a different winding number by approaching from different directions. The cases where it fails seem to be the ones where the point is enclosed by an intersection of the contour. Is the method known to fail in this case and if so, why?

If you have a picture of a contour you can find the winding number by using the rule 'add one if you cross the contour while it's going down' and 'subtract one if you cross it when it's going up'. In some cases however it seems I can get a different winding number by approaching from different directions. The cases where it fails seem to be the ones where the point is enclosed by an intersection of the contour. Is the method known to fail in this case and if so, why?