Find a method to determine whether or not any given knot is invertible.

This is a picture of a knot. For us, a knot is a closed loop of string in 3-dimensional space. Suppose you take a knot and start marking points along the string with consecutive numbers; to do this you have to choose a direction in which to travel along the knot. This choice of a direction is called an orientation of the knot. A knot is invertible if you can smoothly deform it (move it around in space without cutting or breaking it) to look like itself but with the opposite orientation. In other words, if you marked points along the knot in order with the numbers 1,2,3,4,5, then you could smoothly deform the knot so that it looked just like it did when you started, with the point marked 1 at the same point it started at, but with 2 where 5 was, 3 where 4 was, 4 where 3 was, and 5 where 2 was.

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.