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?



The singularity type of a non-torus link
This doesn't answer your more specific request for a construction involving $x^p+y^q=0$, but Benjamin Bode and Mark Dennis have recently given an algorithm for finding "semiholomorphic" complex functions $f:\mathbb{C}^2\rightarrow\mathbb{C}$ (polynomials in $u,v,\bar{v}$ where $\bar{v}$ is the complex conjugate of $v$) such that the intersection of their zero sets with the unit 3-sphere yield any desired braid closure. 


See [these](https://arxiv.org/abs/1612.06801) [two](https://arxiv.org/abs/1612.06328) papers.


Note that the case where the knot can be represented as the link of an isolated singularity of a *complex polynomial* is classical; see the book of Milnor, "Singular points of complex hypersurfaces" or other references cited in the intro to Bode and Dennis's papers.


