# Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:

• A non-trivial connected sum $$M_1\# M_2$$ admits a geometric structure if and only if $$M_1=\mathbb{R}P^3=M_2$$.
• "Most $$3$$-manifolds admit a hyperbolic structure.
• All other $$3$$-manifolds admit one of the other geometric structures or can be cut along $$2$$-spheres and $$2$$-tori into pices that admit geometric structures.

From this it follows that a "randomly chosen" $$3$$-manifold is not a connected sum.

I thought one way to make this precise is to use the hyperbolic Dehn surgery theorem:

• Every $$3$$-manifold can be obtained by Dehn surgery along a link in $$S^3$$.

• If the link is a hyperbolic link, then the resulting manifold is hyperbolic except for only finitly many surgery coefficients.

• A knot in $$S^3$$ is hyperbolic if and only if it is not a torus or a satellite knot.

• On Wikipedia it is written: "Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco."

I have the following questions:

In which ways one can make claims like "a random knot is hyperbolic" precise?

Is it true that a random link is hyperbolic? (Again how to make this precise?)

There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really satisfying) model of three-manifolds is the Dunfield-Thurston model:

Finite covers of random 3-manifolds.
Nathan M. Dunfield, William P. Thurston.
Invent. Math. 166 (2006) 457-521


and a random manifold in this model was proved hyperbolic by Joseph Maher. Lots of related results (and models) can be found in my preprint

Rivin, Igor.
"Statistics of Random 3-Manifolds occasionally fibering over the circle."
arXiv preprint arXiv:1401.5736 (2014).


There are no hyperbolic split links (they are not irreducible)

It is not clear that a random knot is hyperbolic in all models. In my favorite model of random knots, there is a finite probability that the knot is the trefoil. I am not sure that there is any popular model of links, but it is pretty clear that in any reasonable model a random link will not be split.

• Are you sure about the first reference? I would have given the following : Finite covers of random 3-manifolds. Nathan M. Dunfield, William P. Thurston. Invent. Math. 166 (2006) 457-521. (though the two papers are obviously related) – Sylvain Maillot Feb 8 '16 at 17:27
• @SylvainMaillot Oops, canonical error. – Igor Rivin Feb 8 '16 at 17:30
• Igor, what is your "favorite model of random knots that you refer to? – Eric S. Feb 8 '16 at 19:50
• @EricS. My favorite mode is the Fourier model. That is, a knot is a triple of periodic functions from the reals to the reals, so can be represented by Fourier series. So, pick random Fourier series (of course, you can truncare eventually). In order for these to converge to smooth curves, the decay of coefficients has to reasonably rapid (I think $N^{-3/2},$ or thereabouts). If you make that assumption, the distribution will be supported on a finite number of knots, the most probable being the unknot, the trefoil, and the figure eight. – Igor Rivin Feb 8 '16 at 20:09
• @EricS. (interestingly, this is pretty much what is observed in biological systems). – Igor Rivin Feb 8 '16 at 20:10

The answer to the question "what does a random 3-manifold/knot/link look like?" definitely depends on the model. Here are a few references to complement Igor's answer:

This is the paper where Joseph Maher proves that in the Dunfield-Thurston model (based on Heegaard splittings) a typical 3-manifold is hyperbolic:

Joseph Maher. Random Heegaard Splittings http://front.math.ucdavis.edu/0809.4881

Here is a more recent article with applications to non-random questions:

Alexander Lubotzky, Joseph Maher, Conan Wu. Random methods in 3-manifold theory http://front.math.ucdavis.edu/1405.6410

By contrast, there is a (perhaps more naive) model for generating random knots using the Gaussian random walk in Euclidean 3-space. Surprisingly, such a knot is a satellite knot (hence nonhyperbolic) with positive probability:

D. Jungreis. Gaussian random polygons are globally knotted. J. Knot Theory Ramifications 3 (1994), 455–464.

• I did NOT know about Jungreis' paper, thanks! – Igor Rivin Feb 8 '16 at 20:12

As others have indicated, there are many different notions of random 3-manifold or random link. Here are two other types of models of random linking:

1. The Petaluma Model (http://front.math.ucdavis.edu/1411.3308). This model is based on link diagrams with a single multicrossing, with the randomness given by a choice of random permutation, which determines the heights of the arcs relative to one another. They're actually able to compute the distribution of (appropriately scaled) linking numbers on the nose with this model. I don't think that this has been done with any other model. They do some computations of some of the moments of other low order finite type invariants too.

2. The random projection model (http://front.math.ucdavis.edu/1602.01484). This model starts with a fixed embedding of some circles in some high-dimensional Hilbert space, and randomly projects these onto a 3-dimensional subspace. In principal, the moments of the linking numbers ought to be computable. This is an intriguing model because there are continuously many parameters. It's possible that by varying the initial embeddings, these models can be made to limit to other types of models. Maybe that could explain some of the universality observed experimentally and discussed in the Petaluma paper.

As far as I'm aware, no one has examined hyperbolicity in either of these models.