Hairy ball theorem: Do the poles have to be opposite each other?

Question

Many graphical examples of the hairy ball theorem show the cowlicks at opposite poles. Is this arrangement necessary, or arbitrary?

I believe technically the theorem says how many cowlicks there must be, not where they are. Intuitively, it seems like potentially the cowlicks could be moved slightly from the axis. However the extreme case, where both cowlicks are right next to each other, seems difficult to imagine.

Also, if you could move the cowlicks to one side, and the cut out that part of the sphere, you would end up with an almost sphere that has no cowlicks. However, it seems like there must be at least one cowlick even in a partial sphere (if it's more than a hemisphere at least).

Answer 1

It is not necessary. For an extreme counterexample, see this image from the Wikipedia article for the hairy ball theorem:

This is a picture of a hairy ball where the hairs only vanish in one point — but at the cost of making the behavior in a neighborhood of that point more complicated.

Answer 2

The arrangement is arbitrary; the "cowlicks" need not be on opposite poles. If you want a nice step-by-step proof/exercise, see Pugh's Book Real Mathematical Analysis.

Moreover, there need not be more than one cowlick in the statement of the assertion - see wikipedia for instance.