# Piecewise linear vs smooth high dimensional knots

A knot to me is the image of a smooth (pl locally flat) embedding $$S^n \to S^m$$ under the equivalence of smooth (ambient pl) isotopies. And more generally embeddings of closed manifolds $$N \to M$$.

These (pl vs smooth) objects are different in codimension $$>2$$: Haefliger knots are non-trivial while any pl-knot is trivial in this codimension. What about the case of $$m-n=2$$?

Triangulating smooth knots and isotopies seems to be doable, but can we smooth out pl knots and isotopies at least for $$m=4, n=2$$? Is there a reference?