As far as I know, there is no deep significance to this question, but I've been playing around with it for a bit and it seems interesting:

Fix a complex number $c$, and consider the map $J_c: \mathbb{C}\rightarrow\mathbb{C}: x\mapsto x^2+c$. For $x\in\mathbb{C}$ we get a sequence of points $S_x=(x_0=x, x_1=J_c(x_0), . . . , x_{i+1}=J_c(x_i), . . .)$.

Now draw a line segment connecting each $x_i$ to $x_{i+1}$. For almost every choice of $x$, we never have 3 line segments intersect simultaneously; so for such an $x$ we may turn this piecewise linear curve into something resembling a knot diagram, by alternating crossing types: over, under, over, under, . . .

Suppose $x$ is such that this curve only has finitely many crossings. Then we can associate to it a genuine knot diagram in a natural way (this is a far stronger assumption than is needed, but I’m taking it now to make the problem simpler - and the knots tamer): glue $x$ to $x_n$ for some sufficiently large (=past all crossings) $n$.

My question is: what tame knots can we get this way (up to ambient isotopy of course)?