Why is it so difficult to untangle a knot?
Short answer: If there were a simple formula to follow there would also be one for proving or disproving any mathematical statement. Before setting out a sketch of a proof I need to be more clear about the problem. I want to rule out the trivial method of untangling where you simply thread one end backwards through the tangle till it meets the other end. So we're going to imagine it's a loop of string that's knotted, and "untangling" means getting back to a simple loop, without magically passing one bit through another. For most closed loops it's not possible to untangle to a loop. For example if you look at the Trefoil Knot above (a.k.a. Granny Knot) it's obvious that no amount of fiddling with it is going to get rid of the knotty bit and leave just a loop. But if your loop $L$ can be deformed into a simple loop then it follows that the complement in 3D space $\mathbb{R}^3 - L$ can be deformed continuously into the complement of the simple loop $\