Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-1}\}$. Let $R_{2n,l}^{pl}$ be the plat closure of $R_{2n,l}$.

Random knots will be equivalent to other random knots by a sequence of Reidemeister moves.

For a fixed $n$ and large fixed $l$, what is the distribution of the random knots $R_{2n,l}^{pl}$ over their equivalence?

I suspect that the distribution is uniform over knots which have the same number of (minimal) crossings, and that the probability of a knot with lower minimal crossings is higher.