# ambient isotopy and isotopy on knot

this is elementary question about classical knot equivalence.

I know that just isotopy which need not to be ambient is not proper to define knot equivalence because bachelor's unknotting.

but this example is unsatisfied to me, because this unknotting isotopy is not differentiable.

If the isotopy is given "smooth" condition, it seems to define knot equivalence intuitively. i.e $F(x,t): S^1 \times [0,1] \to S^3$ s.t $F(x,t)$ is smooth map and level restriction $F(x,t_0)$ is embedding. Can it define knot equivalence?

I think it may be false.

Any reference is grateful to me.

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The answer to your question is the "isotopy extension theorem". Take a look at Hirsch's "Differential Topology" textbook. – Ryan Budney Sep 7 '10 at 2:55
Ryan, thank you for your kind response to my stupid question. Few minute ago, I found exact answering article about it. The American Mathematical Monthly Vol. 112, No. 5 (May, 2005), pp. 417-425 <Florian Deloup> The Fundamental Group of the Circle Is Trivial – Seonhwa Kim Sep 7 '10 at 8:48