As I mentioned in Spheres – 1 I was momentarily sniped by the idea of knot cobordisms. I’ve downloaded a few papers and will stow those away for later but I want to post just a bit more before I leave this topic. First I want to point the reader towards the definition I am thinking of at the Encyclopedia of Mathematics. The idea is very similar to the cobordisms I’ve been posting about. However it’s hard to see how you’re paste a knot onto the end of a cylinder. I’ve made a sketch below…there are obvious gaps. Since the submanifold that’s homeomorphic to the cylinder is in a 4-manifold does it undergo any moves into the fourth direction? Can it self intersect? We can try a weak analogy. A 0-knot is a point and a 1-cylinder is an interval so we can imagine two 0-knots and the cylinder between them. We know we don’t need any extra dimensions for this but let’s pretend this interval is embedded in a 3-space. It could be embedded such that a projection down to Flatland would render the interval self-intersecting.
So what I’m trying to imagine is how to represent the cylinder in a way that self-intersects but such that it reflects actual movement in the fourth direction which allows the knot to be ‘unraveled’ in some sense. Mind you I’m not sure if a trefoil is cobordic to the unknot. I haven’t been able to find out yet.