In introductory knot theory books, authors usually make a choice of smooth knots or piecewise-linear knots. I often find myself wanting to work in the larger setting of piecewise-smooth knots which subsumes both smooth and PL knots. To do this I would need to prove a "piecewise smooth isotopy-extension theorem" in order to show that knots which are equivalent in the piecewise-smooth sense are equivalent in the usual sense that there exists a continuous ambient isotopy or an ambient homemorphism throwing one onto the other:


Suppose $H: S^1 \times I \rightarrow S^3$ is a continuous isotopy and there exist $$0=t\_0 < t\_1 < t\_2 < ... < t\_k = 2\pi$$ such that $H|\_{[t\_i, t\_{i+1}] \times I}$ is $C^\infty$ for all $i$. Then show that $H$ extends to a continuous isotopy of $S^3$, or at the very least there exists a homeomorphism of $S^3$ carrying $H\_0$ to $H\_1$. Feel free to add hypotheses as needed.


One approach to a proof would replace the piecewise smooth knots $H\_0$ and $H\_1$ with smooth knots using continuous ambient isotopies of $S^3$ (I can do this). Then if I knew that every piecewise smooth isotopy (defined as above) of *smooth* knots can be replaced by a smooth isotopy, I could use the smooth isotopy extension theorem to get a continuous ambient isotopy from $H\_0$ to $H\_1$.


Any thoughts? 



Piecewise Smooth Knot Theory
Everything that you wish for is true.
You can approximate "ambient" (in the sense explained below)
isotopies by smooth isotopies and keep the ends (the knots) fixed or not. This is spelled out in great detail in:


* MR0674117, Bröcker, Theodor; Jänich, Klaus: Introduction to differential topology. Translated from the German by C. B. Thomas and M. J. Thomas. Cambridge University Press, Cambridge-New York, 1982. vii+160 pp.


Also the book [Hirsch: Differential topology] has many details.


Edit:
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Sorry, I looked up ambient isotopy in Wikipedia just now, and it means something different than what I remember. I meant an isotopy which can be extended to an isotopy of tubular neighborhoods. The Wikipedia notion was called diffeotopy, or something like that, was it not?
Of course it gives a finer equivalence relation on closed curves (embedded with a tubular neighborhood).


