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This question on physics stackexchange https://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum.

Given a Brownian motion for time t, link the ends to infinity by horizontal lines parallel to the x-axis, going in opposite directions. The walk will not intersect those lines generically, since a 2d random walk is marginally recurrent. You have then closed a loop on the one-point sphere compactification, and it makes sense to ask what knot you made.

There is a scaling problem, so that the knot you get might be very wild. But one can fix this by asking the right question in the limit. Approximate the Brownian motion with small randomly oriented straight line segments. Then, for long walks, the resulting knot will have a prime decomposition, and it is is very plausible to me that the number of prime knots of each type in the prime decomposition will converge to a fixed distribution in the limit of long walks.

Does this distribution exist?

Is there a more efficient method than simulation to get the distribution?

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    $\begingroup$ I don't see why the count of any prime knot in the decomposition would converge. There is a chance that the next $5$ line segments will add a trefoil which will be unlikely to be undone, so with a million line segments it would be a surprise not to see many $5$-segment trefoils. I expect the count of each knot to diverge to infinity. The growth rate for each knot and the relative growth rates seem interesting. $\endgroup$
    – Douglas Zare
    Commented Aug 15, 2011 at 17:46
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    $\begingroup$ Obviously the count diverges, the question is whether the divergence is linear with a coefficient that depends only on the knot type. $\endgroup$
    – Ron Maimon
    Commented Aug 15, 2011 at 18:32

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I am not sure how your 2D random walk relates to knots but physicists have investigated random knotting in 3D. You may be aware of this already. I learnt about this when the following paper was presented at a meeting in Warsaw.

MR1634449 (99e:57010) Deguchi, Tetsuo ; Tsurusaki, Kyoichi .

Numerical application of knot invariants and universality of random knotting. Knot theory (Warsaw, 1995), 77--85, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998.

The hypothesis is that if you have a model for random knots of length L then the probability that you realise a fixed knot K scales as

$C(K) (L/N)^{m(K)} \exp(-L/N)$

where $C(K)$ depends on the knot and the model, $m(K)$ depends on the knot but not on the model and $N$ depends on the model and not on the knot.

This paper also conjectures that $m(K)$ is additive under connected sum.

I don't know what has happened in this area since then. If this hypothesis is correct then $m(K)$ becomes a fascinating knot invariant.

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  • $\begingroup$ I didn't know, but I probably should have. Thank you. $\endgroup$
    – Ron Maimon
    Commented Aug 15, 2011 at 18:42
  • $\begingroup$ The walk I was thinking about is 3d. The 2d nonrecurrence is just used to close the loop. $\endgroup$
    – Ron Maimon
    Commented Aug 15, 2011 at 18:50

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