Let $X$ be the variety in $\mathbb{C}^2$ defined by the equation $y^2 = x^3$ and let $S_{\varepsilon}$ be the 3-sphere whose equation is $|x|^2 + |y|^2=\varepsilon$. Then in "Singular Points of Complex Hypersurfaces", Milnor quotes Brauner (page 4) who says that $X\cap S_{\varepsilon}$ is a torus knot of type $(3,2)$, i.e. a trefoil knot. A similar question also appears at this page.
What confuses me is that a "knot" is only a meaningful concept in $\mathbb{R}^3$ (i.e. one needs an embedding of $S^1\to \mathbb{R}^3$). Since $X$ is a variety in $\mathbb{C}^2$ we can transform it into a subset of $\mathbb{R}^4$, but then all knot theory is trivial in 4-dimensional space. So I assume that what Brauner means is that since $S_{\varepsilon}$ is a 3-sphere, it is sorta like a ball in $\mathbb{R}^3$. How do we make this precise?
Let us say that we instead have a variety $Y$ with a node singularity $xy = 0$. As before we can intersect $Y$ with the sphere $S_{1}$. It is easy to see that this intersection is parametrized by the points of the form $(e^{i\theta},0)$ and $(0,e^{i\phi})$ as $0\leq \theta,\phi\leq 2\pi$. Therefore, $Y\cap S_1$ is a union of two disjoint circles. But how do we know if it is a link or an unlink? Furthermore, what is the natural embedding here into $\mathbb{R}^3$ so that the link/unlink notion makes sense?
