I imagine many readers of this blog are familiar with the fact that you can knot a circle in 3-space, but not in 4-space. If you enjoy thinking about why that is true, please read on!
Think of euclidean 3-space, as a linear subspace of euclidean 4-space, . So if you have a knotted circle in 3-space, you can consider it as an embedded circle in 4-space. And you can unknot it! I think one of the simplest explanations of of this would be the idea to push the knot up into the 4-th dimension every time a strand is close to being an overcrossing (in a planar diagram). At this stage you could in effect change the crossing to be anything you want, after you’re done modifying the crossings, you could push the knot back into 3-space to get a different knot.
But there’s a more uniform version of this construction. I think I first learned of it from Rolfsen’s textbook. It sits most naturally in a slightly different formalism.
is a smooth embedding and whenever .
is called the space of long knots in . There is a natural inclusion map . I claim it is a null homotopic map.
The idea is pretty simple. When you perform the inclusion, there is a new direction in orthogonal to the original . Let’s call the unit vector in that direction . The idea is to take a little bump function , and add to . Choose so that it is strictly increasing along the interval , and then have it decrease to zero quickly near .
There is a straight-line isotopy from to , and also from to where is the standard inclusion, i.e. always. Similarly there is a straight-line isotopy from to . Assemble these three maps together and you have your null-homotopy of .
Once you see this null-homotopy, notice that there actually two such null-homotopies! If instead of choosing to be increasing along , we could have chosen it to be decreasing along . If you assemble those two null-homotopies together, you get a map , or equivalently .
My question for people here is, is this map null-homotopic?
I don’t know the answer to this question. From the perspective of the Vassiliev spectral sequence, or the Goodwillie embedding calculus, this is a difficult map to understand. You can check that this map is zero on rational homology and rational homotopy groups. Not enough is known about the torsion in homotopy or homology to say what’s going on there — in a way that’s part of why I love this question. It’s also possible one of you will notice there’s a naive null-homotopy of this map. I just don’t see it.
Edit: One further note. There can’t be too naive a null-homotopy of the map . This is because if you evaluate your knots at a finite collection of points in , this map restricts to the “Freudenthal suspension map” . This map is known to be an isomorphism on the first non-trivial homology (and homotopy) group (see for example Fred Cohen’s dissertation). So in particular, given a finite set of points in there are restriction maps . These maps to the configuration spaces are themselves null-homotopic, so they do not inform on whether or not is null-homotopic, but they do put restrictions on the nature of any such null-homotopy, were it to exist.