It's been known for a while that primes in number fields can be thought of, from an algebraic point of view, to be similar to knots in 3-manifolds. A good reference (thanks to this question) would be an article by Morishita, 0904.3399.

There are therefore many good analogues of operations, such as covers, or objects, like zeta-functions, that are defined purely algebraically. For example, a *linking number* of two knots has an easy algebraic definition as the image of one knot in the homology of the complement to the other which is analogous to *residue symbol* in number theory.

However, the operations of taking connected sum and cutting/gluing along a subsurface don't appear immediately to have an analogue in number fields. If you know how to make sense of "gluing" two schemes $\operatorname{Spec} \mathcal{O}_K$ and $\operatorname{Spec} \mathcal{O}_L$ along the "common element $x \in K, L$, by all means, please tell us!

Either way, here's my question:

What could be an analogue of the Thurston geometrization program for number fields?

(may be this analogue will not be using gluing-like operations after all?)