TITLE: Random links and $3$-manifolds
QUESTION [11 upvotes]: 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?
Are there hyperbolic split links?
Is it true that a random link is hyperbolic? (Again how to make this precise?)
Is a random link a split link?

REPLY [6 votes]: 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:

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.
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.