I'm trying to show that we can embed the Klein bottle in $\mathbb{R}^4$. I've previously shown that a figure 8 curve can be embedded in $\mathbb{R}^3$ by a bump function that pushes away the intersection at the origin.
Now I want to use this to embed $K$. I'm aware of the "figure 8" description of $K$ in $\mathbb{R}^3$ where you rotate a figure 8 band while twisting it around the origin, given by:
$$x = (r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v) \cos \theta\\ y =(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v) \sin \theta\\ z = \sin\frac{\theta}{2}\sin v + \cos\frac{\theta}{2}\sin 2v$$
And then I want to push away the figure 8 intersection into $\mathbb{R}^4$. My issue is, how do I show that the above description actually is a Klein bottle?
Thanks