Well, it's discussing having some larger ambient manifold and then embedding other manifolds in it of codimension 2 (and then also taking the complement and stu

Well, it's discussing having some larger ambient manifold and then embedding other manifolds in it of codimension 2 (and then also taking the complement and studying the fundamental group of said complement). If the ambient manifold has dimension 3 instead of 4, that's knot theory. Perhaps there's some reason why that's not the appropriate analogue, but it's certainly the naive analogue; if there's some reason it's not actually the right analogue, you're going to have to explain it. And, again, the point isn't just to have an analogue that one can visualize, but a simpler case where the results are known so that one can compare the results and see how the dimension affects things.