Find a method to determine whether or not any given knot is invertible.
The knot in this picture is not invertible. Click on the image to see another knot that is invertible and to see why it is invertible. If someone gives you a drawing of a knot, can you decide whether or not it is invertible? Can you come up with a general method to do this that works for all knots?
Update: Actually, we were wrong, this problem is solved. Thanks to Ian Agol for explaining this to us, who says that the known algorithm is not necessarily practical (so still an interesting question to find a general practical algorithm), and that for the special case of a hyperbolic knot, there is a computer program called SnapPea that can decide the question pretty easily.