probabilistic knot theory - MathOverflow

probabilistic knot theory

2009-10-30T04:54:27Z

Take a smooth closed curve in the plane. At each self-intersection, randomly choose one of the two pieces and lift it up just out of the plane. (Perturb the curve so there are no triple intersections.) I don't really know anything about knot theory, so I don't even know if I'm asking the right questions here, but I'm wondering: What is the probability that this is the trivial knot? What can we say about how knotted this knot might be, and with what probabilities? (Measure "knottedness" in whatever way you like.) More generally, can we say anything about the probability of the various possible values in the usual invariants that people use to study knots?

I only have an idea of how to approach the first question, and even then it's only by brute force. I was just playing around with the easiest cases, and I think that with 0, 1, or 2 intersections, all knots are trivial, and with 3 intersections the knot is trivial with probability 75%.

A general analysis should presumably involve calculating the probability that we can simplify using various Reidemeister moves, but I don't know how to incorporate this. I'd imagine a computer could brute-force the first few cases pretty easily (I'm not so bold as to venture an order-of-magnitude guess on whether it's the first few hundred or the first few million)...

Answer by Andy Putman for probabilistic knot theory

2009-10-30T05:10:37Z

I suspect that answering this question would be very difficult. A more reasonable question would be to try to understand the distribution of the various numerical knot invariants. I don't know any references off hand, but I know I've heard talks on the subject.

If you want to try to make conjectures about this kind of thing, then I highly recommend Livingston's table of knot invariants, which contains an amazing amount of data.

Answer by Alon Amit for probabilistic knot theory

2009-10-30T05:22:13Z

The model you propose for random knots obviously depends on the curve you draw initially, so I'm not sure this is the most natural model to consider. People have certainly looked at various probability distributions of (various classes of) knots (or knot projections). One of the immediate problems is that even just doing computer simulations is hard since determining the knot type - or just unkottedness - of a given knot diagram is highly non-trivial.

A paper which does this with Vassiliev Invariants (a certain important class of polynomial-like invariants of knots) appears in the volume "Random Knotting and Linking", edited by Millett and Summers (look at the paper by Deguchi and Tsurusaki). Other papers in this volume may interest you, too.

To the best of my knowledge, there's no really model of random knots for which the question "what is the probability that the knot is trivial" has a known answer, except that as the number of crossing tends to infinity this probability likely approaches 0 (as anyone who left a set of mobile headphones in his pocket for more than five minutes knows).

Answer by Scott Morrison for probabilistic knot theory

2009-10-30T06:00:53Z

You should look at the Knot Atlas, which contains lots of tabulated knot invariants, although often not in as convenient form as Livingston's site.

Really, though, you want to download the KnotTheory` package (presupposing you have access to Mathematica), available at the Knot Atlas. With a bit of fiddling, you can easily run experiments of the type you describe. It can calculate many invariants from the presentation of a knot.

Best of all, you should go and think about "physically realistic" models of random knots, and then try to implement such a model using one of the many knot notations the KnotTheory` package understands. There are some good papers written about this subject, and even some real life experiments with strings in boxes being shaken up and down! :-)

Answer by Noah Snyder for probabilistic knot theory

2009-10-30T17:37:40Z

People studying the topology of DNA use various models of random knots. Most of them have some geometric input as DNA has an actual length and doesn't want to bend too much.

Answer by Steve Flammia for probabilistic knot theory

2009-10-30T18:52:20Z

One possible route to a model of random knots would be through the braid group. Every knot can be expressed (non-uniquely) as the closure of a braid. So, for example, you could apply the braid generators uniformly $n$ times across $k$ strands, close the braid using your favorite closure, and then ask this question sensibly. I don't think you can directly ask about the $n \to \infty$ limit for the braid group, though, because I don't think there is a notion of uniform measure for that group. Actually, perhaps I will post this as a separate question, but is the braid group amenable? I would wager that in this model, the probability of having the unknot decreases very quickly with $n$ and $k$.

To test if you have the unknot, it is conjectured that you just have to check the Jones polynomial. But even this is still hard in general, ~~unless~~ even if you happen to have a quantum computer. :)

(Edit: Thanks Greg Kuperberg, below, for the correction.)

Answer by Ryan Budney for probabilistic knot theory

2009-11-04T04:07:39Z

I believe there are a few known "random knotting" type results out there. Not the kind of results the original poster requested, but related. Take n points in R^3 generated by a random walk, join them up (cyclicly) by straight lines. That's generically a knot. And with probability 1 (as n gets large) it's non-trivial and has a trefoil knot summand. The paper by Deguchi and Tsurusaki in "Lectures at Knots '96" provides references for these results although I've never read them in detail.

Answer by Sam Nead for probabilistic knot theory

2009-11-10T03:11:08Z

Just to reply to comments above: if you stick to "random" diagrams with at most say 30 crossings, I am confident that SnapPea will give you answers essentially immediately.

Also, to second suggestions already made, the probabilities you get will depend very sensitively on the model you choose. (Which is why this question is not going to get a real answer!)