Let's take a look at the solutions to the equation:

If we let x and y be complex, the solutions lie in 4 dimensional space (2 for x and 2 for y), and our equation becomes two equations: Real parts are equal and Imaginary parts are equal to each other, so the solution set itself should be 2 dimensional. We can visualize this solution set by taking cross sections of the four dimensional space.



Show Projection
Auto-rotate around conjugate axis
Show hypersphere of radius
a:
b:

Something really interesting happens if we take spherical cross sections around the origin. That is, we take the 3-dimensional surface of a 4-dimensional sphere around the origin and find its intersection with the solution set. We can solve this explicitly to get:

We can then project this intersection to 3 dimensional space via the stereographic projection map:
Use hypersphere of radius
a:
b:
Show Torus

Show Neighborhood
Neighborhood Radius
Show Second-Order Knot
c:
d:

A question we might ask is what knots/links can be obtained by intersecting the solution set of a polynomial equation with the surface of a small 4-dimensional sphere. These knots are the "Compound Cable Knots": (see, for instance, Topology of hypersurface singularities Neumann, Walter D)

One can see that both coordinates in this formula are polynomials in e to a fraction of i θ, and thus have a polynomial relationship.

Our construction above leaves the unit hypersphere, but provided epsilon is small enough, we may deform our way back to the unit hypersphere without changing topology (i.e. crossing a singular point).