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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?

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    $\begingroup$ Not all links of arise from solutions of polynomial equations at all, let alone solutions of this special form. In particular, as you note, this equation produces torus links, so there is no hope to represent non-torus links. $\endgroup$ – Will Sawin Apr 26 '17 at 2:27
  • $\begingroup$ well you can see why I might naively ask the question right? The reason it works for torus links is an action $\mathbb{S}^1\times \mathbb{S}^1$ on the surface, with a particular set of periodicities. Since you can represent any knot as a braid, at least there is a geometric intuition that we might have some hope of doing the same thing in general? $\endgroup$ – cduston Apr 26 '17 at 2:51
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    $\begingroup$ The book of Brieskorn Plane algebraic curves describes 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
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    $\begingroup$ I believe that "very few" knots or links arise in this way (at least from the perspective of knot theorists). The ones which do arise in this way are studied very thoroughly by Eisenbud and Neumann in a book jstor.org/stable/j.ctt1bgzb9w. $\endgroup$ – Peter Samuelson Apr 26 '17 at 13:46
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    $\begingroup$ This article expresses the figure 8 knot as the link of a real singularity: ams.org/mathscinet-getitem?mr=643562 $\endgroup$ – Ian Agol Apr 26 '17 at 16:58
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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 two 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.

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  • $\begingroup$ Yeah I consider this to be the answer. Although it's not in the exact form of the simple polynomial I was looking for, it's a generalized polynomial and you can get any closed braid from it. $\endgroup$ – cduston Apr 28 '17 at 16:19

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