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?
