I recently flew non-stop Washington (IAD) to Beijing (PEK). I expected to start heading in the direction of Alaska, then south over eastern Siberia. Instead, we headed up the eastern Canadian coast and over Greenland towards Russia. When I got home, I got out a length of string and a globe, and saw that the shortest path was almost directly over the north pole (but slightly towards Alaska).

I'm wondering: do planes avoid going over the north pole, even if it's the shortest route? If so, why? In case of emergency landings?
Also, is the usual IAD-PEK route the one my plane took, heading northeast, or is the more usual route a northwest one, and maybe the weather affected the choice of route for my flight.

Anyone know?