Show that two tori that identified $\left(x,y\right)\mapsto\left(y,x\right) $ is homeomorphic to $\mathbb{S}^{3} $

Question

The boundary tori $\mathbb{S}^{1}\times\mathbb{S}^{1}$ of two copies of the solid torus $\mathbb{S}^{1}\times D^{2}$ are identified be a map $\left(x,y\right)\mapsto\left(y,x\right)$. Show that the resulting quotient space is homeomorphic to $\mathbb{S}^3$.

Answer 1

Hint: Consider the map from the disjoint union of two full tori to $\mathbb{C}^2$ given by

$$\begin{gather} \Phi \colon \mathbb{S}^1\times D^2 \times \{1,\,2\} \to \mathbb{C}^2\\ \Phi (z,\,w,\,1) = (z,\,w)\\ \Phi (z,\,w,\,2) = (w,\,z) \end{gather}$$