It is a well-know result that singular surfaces like

$$x^p+y^q=0$$

(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small sphere around the point $(0,0)$. These are called Pham-Brieskorn manifolds.

I'm wondering if there is a general way of representing the singular surface for other knots and links. To try and be more specific, I'll guess an answer, and I'll restrict attention to knots (rather than links).

Representing a knot $K$ as a braid generated by a braid word $\sigma_1\sigma_2...\sigma_i$, can we find a surface

$$x^p+y^q=0$$

which when intersected with a small ball around the origin gives us the braid. If so, how are the integers $p$ and $q$ related to the defining braid word?

Plane algebraic curvesdescribes all the knots that are links of singularities of plane curves. They are all iterated torus knots and Brieskorn explains how to associate an equation to such a knot. $\endgroup$ – Liviu Nicolaescu Apr 26 '17 at 9:37