In Jeffrey Weeks book ["The Shape of Space"](https://books.google.de/books/about/The_Shape_of_Space.html?id=Lurp6nB4LtQC&redir_esc=y) 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](https://en.wikipedia.org/wiki/Hyperbolic_link) 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?
> 
> 
> 



Random links and $3$-manifolds
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.


