I am trying to find two embeddings $f$ and $g$ of the circle $S^{1}$ into a space $X$ such that the subsets $f(S^{1})$ and $g(S^{1})$ are ambient isotopic within $X$, but $f$ and $g$ are not ambient isotopic within $X$.
This is problem 10 in "An Illustrated Introduction to Topology and Homotopy", by Sasho Kalajdzievski - page 226. It is before the material on homotopy and the fundamental group. Hence, I assume none of that is needed to provide the answer.