Suppose there are two topological embedding $f^k: T^n=(S^1)^n\to (\mathbb C^*)^n$ ($k=0,1$). They respectively induce homomorphisms $$F^k: \ \ \pi_1(T^n)\to \pi_1((\mathbb C^*)^n)$$
Is it true that $f^0$ and $f^1$ are isotopic to each other if and only if $F^0=F^1$? How to prove?
EDIT: Here the isotopy means: we can find a continuous family $f^s: T^n\to (\mathbb C^*)^n$ for $s\in[0,1]$ so that every $f^s$ is a topological embedding.
I would like to ask some further questions:
Can we classify all topological embeddings from $T^n$ to $(\mathbb C^*)^n$ up to homotopy/isotopy?
Roughly speaking, I was wondering if there is an exotic $T^n\to(\mathbb C^*)^n$ which is not the `same' as the standard $T^n\subset (\mathbb C^*)^n$