There could be a very hands-down, but still general, approach to this problem, that to a certain extent permits to calculate the distribution of some invariants.
Consider the configuration space of n points in the plane $X_n := Conf_n (\mathbb{R}^2) $: a point here consist of a tuple $x_1, \ldots, x_n$ of distinct points in $\mathbb{R}^2$. The topology here is given by being a subspace of some $\mathbb{R}^N$.
Now to a point in $X_n$ you can associate a closed curve in the plane in the following way: go from $x_i$ to $x_{i+1}$ linearly, and then go back from $x_n$ to $x_1$. This is not smooth, but any smooth immersion sufficiently close to this will do the job; in other words, if you take any lift to the space of this piecewise linear knot, the smoothly perturbed one will lift to a homotopic knot.
You can also go in the other direction: given a closed immersed curve $C$ in the plane, you can define the n-th truncation as the configuration of points of values of the curve at $0, 1/n, 2/n, \ldots, (n-1) /n$. As above, this could be not defined because points could be not distincts; but perturbing slightly some of the values $k/n$ will result in a (homotopic) configuration of points, since the curve is embedded. Maybe one could be even more formal about configurations by speaking directly about configurations obtained in the space by lifts of the curve, but this is heuristics.
The thing is that now we are somehow convinced that $X_n:= Conf_n(\mathbb{R}^2) $ is an approximation of the space of immersed curves, and it is finite dimensional. Since the (homotopic type of the) knot associated to a configuration is invariant by translations and positive dilatations (of both coordinates at the same time), we can equivalent study $Y_n := X_n /\{ translations, dilatations\}$. The cool fact is that now this space is finite dimensional it has a natural probability measure $\mu_n$, because it will be something like a subset of a sphere. Note that the subsets we wanted to measure in $X_n$ (configurations of points such that the associated knot has some homotopy type) are invariant by translations and dilatations, so they descend to subset of $Y_n$ and we can measure them with $\mu_n$.
The process I propose would be the following. Say you want the probability that some lifts of a plane curve as a homotopic invariant property $H$ and call $A_H$ the subsets of curves that has such homotopy type. Set $A^{(n) }_H$ as the subset of $Y_n$ of configurations (up to translations and dilatations) that yields knots in $A_H$. We'd like to set
$$ p( C \text{ has } H) = \lim_{n \to \infty} \mu_n(A^{(n)}_H) $$
Even though this could seem very abstract, there is a possibility that one could compute $\mu_n(A^n_H) $ by induction on $n$. Indeed, the $(n+1) $-th knots are obtained by the $n$-th knots by allowing the last edge to have one "flexibility" point. By inductive hypothesis, you roughly know that the rest of the knot has 'arrangement' $a_1$ with probability $p_1, \ldots,$ arrangement $a_k$ with probability $p_k$. Now you insert this new point that forms a "V" with first and n-th point and mess up somehow the knot, and you sum up this messing up by taking into account the probability that such a new point falls there.
I know that this is very heuristic, but in practice it just means: hey let's suppose that the knots are piecewise linear".
Bye! Let me know what you think. The approximation with configuration spaces in the space is not completely justified, because goodwill calculus only works for dimension greater than 4; but I still think that this could give a cool approximation to the OP problem. If time permits, I'll try to approach the Alexander Polynomial distribution case and see what happens.
