I don't know about the polymath project but here is one thought:

A long knot is an embedding of $\mathbb{R}\rightarrow\mathbb{R}^3$ which as $t$ tends to $\pm\infty$ approach the line $x=y=z$. Examples are given by $t\mapsto (x(t),y(t)z(t))$
where $x(t)$, $y(t)$, $z(t)$ are monic polynomials of degree $2r+1$. In fact all long knots arise this way. However when I have implemented this you get pretty unsatisfactory pictures. The problem is to find a way to get better pictures (not exactly cutting edge research, I know).

One possibility would be to define an energy functional and then take the gradient flow to find a local minimum. If we fix $r$ this all takes place on a finite dimensional manifold.

Another direction is to apply a Mobius transformation that moves the point at infinity. This gives a knot parametrised by rational functions. I haven't tried this, but I doubt it gives a pretty picture. Can these pictures be improved?

You could also investigate this from the point of view of Vassiliev theory (which is how it came up when I heard about it). That is, look at the discriminant, the polynomials whose long knots have self-intersections.