The problem statement:
Exhibit explicit parameterizations covering $S^1 \times S^1 \subset \mathbb{R}^4$.
My question:
I had my attempt below, but my question lays at the very last. It seems my approach is not well defined. Say, given $(r, \theta, z)$ that $z<0, r > 0$, should $\phi$ go to the second one or third one? My guess is that they are the same so it doesn't matter. My concern is that though the two solutions from $z<0$ and $r>0$ coincide, it doesn't literally equivalent to well-defined.
By definition, a parametrization of the neighborhood $V$ is a diffeomorphism $\phi: U \rightarrow V$. Therefore, we look for a function $\phi$ that is a diffeomorphism between some neighborhood $U$ and $S^1 \times S^1$:
Let $T_{a,b}$, and the circle of radius $a$ in $xy$ plane is described by $$C_a = \{(x,y,0) \in \mathbb{R}^3\;:\; x^2 + y^2 = b^2\}.$$ And $$T_{a,b} = \{(x,y,z)\;:\; \text{dist}((x,y,z), C_a) = b\}.$$ In polar system as $(r, \theta, z)$, where $(r, \theta)$ are the polar coordinates in the $xy$ plane, and $z$ is the height: $$C_a = \{(a, \theta, 0)\;:\;\theta \in [0,2\pi)\}.$$ Then $$T_{a,b} = \{(r, \theta, z) \;:\; (r-a)^2 + z^2 = b^2\}.$$ So $$\phi: T_{a.b} \rightarrow S^1 \times S^1$$ is given by $$\phi((r, \theta, z)) = (1, \theta, 1, \eta)$$ where $\eta \in [0, 2\pi).$ The function $\phi$ is injective because if $(1, \theta, 1, \eta) = (1, \theta^\prime, 1, \eta^\prime)$, $\theta = \theta^\prime, \eta = \eta^\prime$. Therefore, $b \cos(\eta) = b \cos(\eta^\prime)$. That means $r-a = r^\prime - a$. So $r = r^\prime$. Similarly, $z = z^\prime$.
\begin{eqnarray} \phi_{z>0}((r,\theta,z)) &=& (1,\theta,1,\arccos(\frac{r-a}{b}))\\ \phi_{z \leq 0}((r,\theta,z)) &=& (1,\theta,1,-\arccos(\frac{r-a}{b}))\\ \phi_{r>0}((r,\theta,z)) &=& (1,\theta,1,\arcsin(\frac{z}{b}))\\ \phi_{r \leq 0}((r,\theta,z)) &=& (1,\theta,1,\pi + \arcsin(\frac{z}{b})) \end{eqnarray}
The neatest parametrization uses complex notation, replacing $\mathbb R^4$ with $\mathbb C^2$, where the torus is represented by $\{(z,w):|z|=|w|=1\}$. $$(t,s)\mapsto (e^{it},e^{is})\in \mathbb C^2$$ In terms of real coordinates, it's $$(t,s)\mapsto (\cos t, \sin t, \cos s, \sin s)\in \mathbb R^4$$