# Every knot is slice, find the error in the following argument

It is well known that exist knots which are not slice, however I came up with the following argument contradicting this fact and I fail to see where the mistake is, can somebody help me?

Let $$K_0$$ and $$K_1$$ be two knots in $$S^3 = \partial D^4$$. Since $$K_0$$ and $$K_1$$ are homotopic in $$D^4$$ they are isotopic in $$D^4$$. By the isotopy extension theorem, we can therefore find an ambient isotopy $$F:D^4\to D^4$$ sending $$K_0$$ to $$K_1$$.

Now suppose that $$K_0$$ is slice with slice disk $$\Delta\subset D^4$$, then we conclude that $$K_1 = \partial F(\Delta)$$. This would imply that every knot is slice.

Evidently your application of the isotopy extension theorem is flawed: every ambient isotopy $$D^4 \to D^4$$ restricts to an ambient isotopy $$S^3 \to S^3$$, so your same proof would show that any two knots in $$S^3$$ are ambient isotopic, which is false.