In Libgober's paper [Alexander polynomial of plane algebraic curves and cyclic multiple planes](http://homepages.math.uic.edu/%7Elibgober/otherpapers/export/1982alexanderduke.pdf), Example 2 (p.850), Libgober claims that the complement to this curve (i.e. $x^2u=y^3$ relative to the line in infinity $u=0$) is a retract of the complement of the trefoil knot in $S^3$. I wonder how to see this. Is in general the complement of a plane curve (relative to a line in infinity) the retract of a knot in $S^3$? If not, when is that true?


I heard that the complement to an affine curve retracts on a complement to a knot (for any weighted homogenous curve). Is that true and is there a reference?



Complement of plane curve and knot
If you have a weighted homogeneous polynomial $f(z\_1,z\_2)$ then that
means there's a $\mathbb{C}^\times$ action which preserves both
the curve $C=\{f=0\}$ and its complement. Take a small sphere
$S$ centred at the origin which is transverse to the orbits of
this group action. Its intersection with the curve is a link $L$
(not necessarily a knot); these knots and links are examples of
algebraic links. They are well-studied and two of the classic
references are Milnor "Singular points of complex hypersurfaces" (Chapter 10)
and Brieskorn--Knoerrer "Plane algebraic curves" (Chapter 8.5). The subset
$S\setminus L\subset\mathbb{C}^2\setminus C$ is a deformation
retract: you can construct the deformation retraction using the
scaling action of $\exp(\mathbb{R})\subset\mathbb{C}^\times$ to
move points into $S$.


The simplest example is the curve $\{z\_1=0\}$, which gives the
unknot in $S^3$. If you take $\{z\_1z\_2=0\}$ then you get the
2-component Hopf link. You can see this by stereographically
projecting the 3-sphere $|z\_1|^2+|z\_2|^2=1$ from the point
$(0,i)$. The intersection of the curve with the sphere consists
of the unit circles in the $z\_1$ and $z\_2$ planes, which project
respectively to the unit circle in the $xy$-plane and the
$z$-axis. When you project instead from a nearby point that
isn't on the $z\_2$-axis, this vertical line becomes a very big
circle that links with the first one.


It's a bit harder to see why you get the trefoil from
$z\_1^2=z\_2^3$. More generally $z\_1^p=z\_2^q$ gives you the
$(p,q)$-torus link. Brieskorn and Knoerrer give a good
explanation of how to figure out what you get. More generally,
an algebraic link is obtained by intersecting a small sphere
with a curve, but the curve doesn't have to be weighted
homogeneous. In that case, you get an "iterated torus knot": you
take a torus knot and thicken it, then stick another torus knot
on the boundary of the thickening, etc. Which torus knots you
use depends on the Newton-Puiseux expansion of $f$. Edit: Of course if you don't have a weighted homogeneous polynomial then you won't be able to say anything global about the complement of the curve; the comment about iterated torus knots only applies to how the curve intersects a small sphere centred at a point on the curve. If you're interested in general curve complements then there are also methods for studying them with a knotty flavour (e.g. braid monodromy) but there's no reason to expect them to retract onto knot complements.


