So they touch on 4-dimensional theories only briefly, but one thing I'm very curious about is if higher dimensional theories provide a richer computational structure, in any of several senses.

Every 1-knot has a sort of 2-knot analog (actually, a whole family of them) made by "revolving" the knot, but there are 2-knots not generated in this manner. (The same holds true for higher dimensional knots.)

As we go higher, there are knot moves not even possible in lower dimensions (such as twists).

Just a thought.