@AkivaWeinberger You're about to ask an interesting question, one that depends on what you think a "knot" is. If knots, to you, are continuous/PL embeddings up to locally flat (or, if you prefer, ambient) isotopy, then knots only exist in codimension 2, and they always exist in codimension 2.
If they're smooth embeddings up to smooth isotopy, the story is more complicated; Thom constructed a smoothly knotted $S^3$ in $S^6$.
I'm mostly convinced the separation of fields is somewhat semantics and personal preference. I have seen people with work which to me clearly labels them "topologists" to me call themselves "geometers" and vice versa.
@PVAL, when I finished my Ph.D. and applied for jobs, U Wisconsin had no "geometry" checkbox, and I had to say I was a topologist (was I going to say logician?). But complex integral geometry, despite being filled with Chern classes, was not topology.