# 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?

## 1 Answer

PL locally flat knots have rank 2 $$PL(2)$$ normal bundles, but $$PL(2)$$ can be reduced to $$SO(2)$$ structure group. So there is a PL embedding of the smooth manifold $$\nu S^2 \cong S^2 \times D^2 \hookrightarrow S^4$$. This induces a smooth structure on a neighbourhood of the knot, namely the image of $$\nu S^2$$. Since the embedding is PL, there is a PL structure on the exterior of the knot $$S^4 \setminus \nu S^2$$, and the PL structures on the intersection $$S^1 \times S^2$$ coming from the exterior and coming from the smooth structure on $$\nu S^2$$ agree. Then use: given a PL structure on a 4-manifold with boundary $$X$$, together with a smooth structure on $$\partial X$$ refining the PL structure, there is (up to isotopy) a unique smooth structure on $$X$$ refining the PL structure (and restricting to the given smooth structure on $$\partial X$$, but this is automatic by uniqueness of smooth structures on 3-manifolds). This extends the smooth structure on $$\nu S^2$$ to all of $$S^4$$. Since this smooth structure on $$S^4$$ refines the standard PL structure on $$S^4$$, it is isotopic to the standard smooth structure on $$S^4$$. This isotopes the knot to a smooth embedding in the standard smooth structure.

Partial attempt at references. PL bundles: Rourke-Sanderson, Kuiper-Lashof. PL $$\Rightarrow$$ smooth for 4-manifolds: the key is Cerf's result $$\Gamma_4=0$$. Main reference is Hirsch-Mazur, Smoothings of Piecewise Linear Manifolds,