$S^1\times S^2$ embedded in $\mathbb{R}^4$

Question

It's easy to embed $S^1\times S^2$ in $\mathbb{R}^5$, since $S^1\subset \mathbb{R}^2$ and $S^2\subset\mathbb{R}^3$, but $S^1\times S^2$ lives also in $\mathbb{R}^4$. How can we write the embedding map $S^1\times S^2\hookrightarrow\mathbb{R}^4$?

Answer 1

Here is an explicit construction: $$(x,y,a,b,c)\mapsto ((2+x)a, (2+x)b, (2+x)c, y)$$ where $x^2+y^2=a^2+b^2+c^2=1$. Note that given

$(\alpha, \beta, \gamma, \delta)$ in the image, we can recover $x=\frac{\alpha^2+\beta^2+\gamma^2+\delta^2-5}{4}$, and hence $a,b,c$.