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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.

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  • $\begingroup$ If each letter is chosen randomly from the braid generators, the closure may not be a knot. If you do not insist that the closure must be a knot. A result in [Jiming Ma, THE CLOSURE OF A RANDOM BRAIDIS A HYPERBOLIC LINK, PAMS, 142(2), 2014, 695-701] says that the probability that the closure is a hyperbolic link converges to 1 when the length $l\rightarrow\infty$. $\endgroup$
    – Zhiyun Cheng
    Commented Jun 11, 2016 at 5:37
  • $\begingroup$ Thank you, I'll have a read of the reference. I was abusing terminology - by knot I mean knots and links. $\endgroup$
    – Ryan
    Commented Jun 11, 2016 at 5:43

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