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)?

  • $\begingroup$ I am confused by your suggestion that you would obtain all knots (i.e. alternating and non-alternating), because aren't you by construction always going to have an alternating diagram for your knot? $\endgroup$ – Neil Hoffman Mar 7 '16 at 4:33
  • $\begingroup$ @NeilHoffman D'oh, that's obvious. Fixed. $\endgroup$ – Noah Schweber Mar 7 '16 at 4:37
  • $\begingroup$ It's not clear to me that you get an alternating knot. The construction is ambiguous to me. Do you choose the crossings with all other parts of the curve to be alternating, or do you just make the crossings with earlier parts of the curve alternate? If the latter, then you don't have to get an alternating knot when you assign crossings along an oriented loop. $\endgroup$ – Douglas Zare Mar 7 '16 at 7:28

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