Knot cobordisms – a brief aside

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.



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